Complete course of self-study I am about $16$ years old and I have just started studying some college mathematics. I may never manage to get into a proper or good university (I do not trust fate) but I want to really study mathematics.
I request people to tell me what topics an undergraduate may/must study and the books that you highly recommend (please do not ask me to define an undergraduate).  
Background:


*

*Single variable calculus from Apostol's book Calculus;

*I have started IN Herstein's topics in algebra;

*I have a limited knowledge of linear algebra: I only know what a basis is, a dimension is, a bit of transpose, inverse of a matrix, determinants defined in terms of co-factors, etc., but no more;

*absolutely elementary point set topology. I think open and closed balls, limit points, compactness, Bolzano-Weirstrass theorem (I may have forgotten this topology bit);

*binomial coefficients, recursions, bijections;

*very elementary number theory: divisibility, modular arithmetic, Fermat's little theorem, Euler's phi function, etc.
I asked a similar question (covering less ground than this one) some time back which received no answers and which I deleted. Even if I do not manage to get into a good university, I wish to self-study mathematics. I thanks all those who help me and all those who give me their valuable criticism and advice.
P.S.: Thanks all of you. Time for me to start studying.
 A: Lang's Undergraduate Analysis is richly illustrated, rigorous, very geometric (vector fields on spheres are discussed) and contains some juicy calculations (look at  the treatment of the Fejer or the heat kernel   ).
Above all it prepares you to advanced modern analysis: Chapter  XIV on the Fourier integral for example starts with a section on Schwartz space (again with some non-trivial calculations thrown in)  which will ease the transition to distribution theory and partial differential equations.
Don't forget that analysis is not  a sterile exercise in axiomatics, despite what some boring books would make you believe,   but one of the most useful and exciting subjects  in mathematics (and physics). 
NB Lang was not an analyst (he was a student of the great arithmetician Emil Artin) but was capable  of using quite tough analysis: just look at his book $SL_2(\mathbb R)$ and you will see Sobolev spaces, Mellin and zeta transforms, resolvants of Laplace operators,... at work.
A: Something that surprisingly has not been said, i find it amazing that at 16 years old you're into math that much and from the list stating your current knowledge, i see that you already acquired some overview on different fields of mathematics, which is great.
Now since there have been some books recommended, i won't go into that anymore. But i would like to give you the following tips:
If you look at the website of a university, often you can see a curriculum of the first few years in math. I'm pretty sure that you can find a university that has detailed course descriptions, including what literature they use. If you look further there might even be course notes online. This not only provides you some of the requested literature, but also can give you a good idea of what essential mathematical knowledge is, treated in the first years at uni. For example, my uni www.uva.nl has course descriptions, and you can select the language to be English.
Also, if you look around and aren't afraid of rejections, i guess that if you email professors at some uni (anywhere around the world), there might be one enthousiastic and willing to help in some way. You might not realize it, but what you're doing is special. Maybe 1 out of 10 or 100 will actually respond, but hey, emailing is free. You might even have a better shot if you dont mail professors but graduate or post-graduate students.
Lastly, considering your enthousiasm and the knowledge you already acquired, you should definitely go for a place at a uni. I have no idea where you live, but i believe some European universities offer grants to foreign students as well (i have NO personal experience with this though), so look around, there might be more possibilities then you think..
Good luck!
Oh and by the way, Singh: Fermat's last theorem is a great popular math book!
A: I do not know in which field you are interested, though it is too early for you to select any field of interest, and rather learn all the basic techniques. Apart from Herstein, you should look at Artin's Algebra. Not only it is lucid, but also it clears several concepts from a practical (applied?) point of view. In particular, you should see the linear algebra portions. Also the exercises are important and should be attempted to make any real progress.
Along with Rudin, you may try to work out Calculus I and II by Apostol. If these are too easy for you, then check differential geometry by Pressley.
Some knowledge in probability theory is always useful. You can try Chung. 
All the best.
A: Linear algebra is the most indispensable subject in all of mathematics: I can easily imagine someone getting a Fields medal without knowing what the sine function is, but I don't believe one could get that prize without being familiar with linear maps.
Linear algebra is an easy subject and the main difficulty is choosing between the thousands of books on the subject and even between the tens of excellent ones.                   
1) My initiation was through Lang's Linear Algebra but I cannot guarantee that my genuine enthusiasm for that book is not  nostalgia-tinted . 
2) Lipschutz-Lipson's book in the Schaum series is very elementary, pleasant and richly illustrated, as befits the subject.
As in all books in the series, the theory is kept to a minimum and the reader is encouraged to recreate the subject by solving judicious exercises (provided with complete solutions, just in case!)    
3) Another excellent classic is Hoffman's-Kunze's Linear Algebra , which is very solid albeit a little  austere. It is more advanced than the preceding two.  
A: I highly recommend The Princeton Companion to Mathematics - an encyclopedic overview of pure math and some theoretical physics with chapters on proof, many areas of math and biographies of famous mathematicians.Math blogs and personal websites. It is great for getting motivation and an overview on most subject areas so you can pick which to study further. It lists further reading for most topics. You should be able to read at your local library.
As well as book there are some great math blogs, here are 3 that are a good start.


*

*Tim Gowers - lots of interesting articles and solved problems
http://gowers.wordpress.com/ 

*Terence Tao - great discussion of topics, open problems and career advice http://terrytao.wordpress.com/

*Tom Körner - learning guides, lecture notes and more http://www.dpmms.cam.ac.uk/~twk/
Any many universities put course materials online so that you can both see what topics math undergraduates study and read the materials and problem sets for free.


*

*Cambridge - Pure math example sheets, lecture notes http://www.maths.cam.ac.uk/undergrad/course/  (drill down into the staff person pages https://www.dpmms.cam.ac.uk/people/ to see posted lecture notes)

*MIT - math lecture notes and example sheets. Some are full OCW Scholar courses that are designed for independent learners who have few additional resources available to them. These courses include exam solution notes, online study groups, video and simulations. http://ocw.mit.edu/courses/mathematics/

*Open University - OpenLearn free course materials on pure, applied and statistics http://openlearn.open.ac.uk/course/category.php?id=8

*Udacity - free video lectures, online tests and learning community mainly related to applied, applicable and computer science topics. http://www.udacity.com/
A: On top of all helpful answers above I can add one very popular book: $\mathit{Concrete \ Mathematics}$ by Graham, Knuth and Patashnik (1995 edition). The main reasons are:


*

*This book is aimed at Computer Scientists that consider (or want to consider) themselves mathematicians through better understanding of mathematics behind programming and algorithms

*It has heaps of awesome problems in areas usually not very well covered by Discrete Math books: Generating functions, series, probability, asymptotics .

*Complexity of problems varies from easy high-school till PhD/open problems. All have an answer or hints (not just 'odd-numbered').

*It blends continuous and discrete (hence 'concrete') math.

*It focuses a lot on recurrences, from very simple to very complicated.
A: This is a recapitulation and extension of what we talked about in chat.
Whatever you do, I recommend that you try a variety of areas in order to find out what you like best. Don’t feel obliged to stick to the most common ones, either; for instance, if you find that you’ve a taste for set theory, give it a try. 
My own interests are outside the undergraduate mainstream, so in mainstream areas others can probably give better recommendations. I do know that you’re working through Herstein for algebra; although it’s a little old-fashioned, it’s still a fine book, and anyone who can do the harder problems in it is doing well. 
You mentioned that you’d prefer books and notes that are freely available. The revised version of Judy Roitman’s Introduction to Modern Set Theory is pretty good and is available here as a PDF. You can also get it from Barnes & Noble for $8.99. Introduction to Set Theory by Hrbacek & Jech is also good, but it’s not freely available (or at least not legitimately so). 
I’ve not seen a freely available topology text that I like; in particular, I’m not fond of Morris, Topology Without Tears, though I’ve certainly seen worse. If you’re willing to spend a little and like the idea of a book that proves only the hardest results and leaves the rest to the reader, you could do a lot worse than John Greever’s Theory and Examples of Point-Set Topology. It’s out of print, but Amazon has several very inexpensive used copies. (This book was designed for use in a course taught using the so-called Moore method. It’s excellent for self-study if you have someone available to answer questions if you get stuck, but SE offers exactly that. In the interests of full disclosure I should probably mention that I first learned topology from this book when it was still mimeographed typescript.) If I were to pick a single undergraduate topology book to serve both as a text and a reference, it would probably be Topology, by James Munkres, but I don’t believe that it’s (legitimately) freely available. You might instead consider Stephen Willard, General Topology; it’s at a very slightly higher level than the Munkres, but it’s also well-written, and the Dover edition is very inexpensive.
I can’t speak to its quality, but Robert B. Ash has a first-year graduate algebra text available here; it includes solutions to the exercises, and it introduces some topics not touched by Herstein. He has some other texts available from this page; the algebra ones are more advanced graduate level texts, but the complex analysis text requires only a basic real analysis or advanced calculus course.
This page has links to quite a collection of freely available math books, including several real analysis texts; I’ve not looked at them, so I can’t make any very confident recommendations, but if nothing else there may be some useful ancillary texts there. I will say that this analysis text by Elias Zakon and the companion second volume look pretty decent at first glance. For that matter, the intermediate-level book on number theory by Leo Moser available here looks pretty good, too, apart from having very few exercises. Oh, come to think of it there is one real analysis book that I want to mention: DePree and Swartz, Introduction to Real Analysis, if only for its wonderful introduction to the gauge integral.
A: When I was your age doing what you are doing, my Analysis professor said I should read Irving Kaplansky's book Set Theory and Metric Spaces.  This was good advice. This book was my introduction to set theory, and ideas like well-orderings whose ideas underlie  much of modern mathematics. I found it very readable and enjoyed it; it was an excellent supplement to Rudin, which others in this thread have recommended.
A: [Not a direct answer but may help in conjunction with the other answers that give topics and subjects of study]
If you've not already checked it out then take a look at MITs "OpenCourseWare" site - I used this to brush up on my linear algebra when getting back into study around 3 years ago.  As well as a wealth of lectures in many subjects the site also provides many, many references, including some that are out of print.  If you pick through the courses carefully you will also find a number of assignments (with answers).
http://ocw.mit.edu/courses/mathematics/
Full disclosure: I am not associated with MIT or the OCW site in any way, I am just a (very) happy user of their site!
A: You can use this link to get Great Stuff like: free video lectures (from top Indian professors) along with Lecture Notes and Good References:
http://nptel.iitm.ac.in/ (Select "Mathematics" from the list of courses available).
A: you must see this:
Chicago undergraduate mathematics bibliography
great programme to follow.
A: I think the following subjects are absolute minimum.
IMO, you should learn those before you learn other subjects(PDE, algebraic topology, differential geometry, algebraic geometry, etc.).
Perhaps other people will recommend good books on each subject.
- Linear algebra
- Calculus(single and multivariable)
- General topology
- Abstract algebra(basics)
- Naive set theory(basics)
A: Here's a bunch of lesser known material I found really good. It may be a somewhat biased list though, since I tend to like reading "easy" books first to get the main idea, then solving hard problems and moving on to more difficult books later. So, some of this list is less textbook'y and more motivational. It's also a bit biased towards mathematics that is relevant to numerical methods, as that is field of personal interest to me.
Linear Algebra:

*

*Gilbert Strang's video lectures
Numerical Linear Algebra:

*

*Trefethen & Bau's book
General techniques for solving mathematical problems:

*

*How to Solve It

*Terry Tao's book
Abstract Algebra:

*

*Benedict Gross's video lectures

*Pinter's Book
Convex analysis and optimization:

*

*Steven Boyd's book

*Steven Boyd's video lectures

*Numerical Optimization by Nocedal and Wright
Real Analysis:

*

*Francis Su's video lectures
Functional Analysis:

*

*Kreyszig's book
Mathematical inequalities:

*

*The Cauchy Schwarz Master Class
Overview of mathematics from Courant:

*

*What is Mathematics
Topology:

*

*Munkres book
Mathematical physics:

*

*Vladimir Arnold's book

*Road to Reality
Numerical/computational mathematics

*

*Gilbert Strang's video lectures on computational science

*Gilbert Strang's video lectures on mathematical methods

*Wolfgang Bangerth's video lectures on finite element methods
Differential topology/geometry:

*

*The Shape of Space

*Frederic Schuller's video lectures
Dynamical systems:

*

*Steven Strogatz book

*Steven Strogatz video lectures
A: I would begin with the following:
1) Everyone has to learn multivariable calculus. There are many books to choose from here. The one I liked best was Calculus: A Complete Course by Adams.
2) You should be comfortable with linear algebra. I see that you know something about it, but you should really learn more. I'm not really sure which book you should use here (I used Elementary Linear Algebra by Edwards and Penney which was ok). Any suggestions?
Then you can learn more advanced mathematics like the text by Rudin which user37787 recommended. For topology I would recommend Topology by Munkres.
You could also look into applied mathematics like statistics and numerical mathematics. I don't know much about these fields so I cannot recommend anything in particular here.
A: As many have said, Rudin's 'Principle of Mathematical Analysis' is a classic for Analysis. My personal recommendation for point-set topology would be Sutherland's 'Introduction to Metric and Topological Spaces'. It basically builds on from what we know in Analysis into more general spaces and the proofs inside are quite neat in my opinion. 
Apart from studying textbooks alone, have you considered reading books on mathematics itself? Books such as 'What is Mathematics' by Stewart and Courant, 'A very short introduction to Mathematics' by Tim Gowers are must reads, and I would also recommend 'The mathematical experience' by Hersh and Davis for a more philosophical insight.
Happy reading!
A: Principles of Mathematical Analysis by Rudin is strongly recommended. 
Although it may be difficult for you, it will be very impressive if you can read through it. Although I don't know your interests, this book is critical for all students studying mathematics. 
A: This question is quite interesting and possibly deserves a big list of answers, so I would add my point of view. This is just an opinion.
First of all, the goals should be set much clearer that it is done in the current version of the question. Even when studying at a proper University one needs to be very specific on what he or she wants to achieve. Examples are: "I want to prepare for a course of General Relativity", or "I want to improve my mathematics for a Computer Graphics project", or "I want to contribute to ... and this is going to be my hobby" etc.
Secondly, I would recommend to find a challenging book that is not a leisure reading but rather a masterpiece in the area that you want to make your own. For instance, you could find one of S.Tabachnikov's books here as a motivating conundrum to start with. Just a hint. no more.
Then, trying to work through the book you will encounter the parts of the story that are blurry or confusing. Thus you will get a good reason to learn more, and more and more. This is perhaps the way how to learn to ask right questions. (I believe that mathematics is all about that, and I feel that I still have to improve my question asking skills...)
Of course, you will need to learn the cornerstones. Try to learn as much Linear Algebra as you can (this includes multilinear algebra). This will help you to master multivariable calculus to the level when it is done on smooth manifolds. A good knowledge of combinatorics is often an advantage and helps to master things like representation theory. My personal preference is geometry that is an enormous area but a course on differential geometry of curves and surfaces would dramatically expand the dimensionality of one's perception of mathematical problems.
A: The Foundations of Mathematics by
Ian Stewart and David Tall (available second hand at Abe books). This is a great book on a variety of topics before "more serious" study, but considering what you say in you have already been reading it may be on the easy side, nevertheless I think it is worthy of a read.
Definitely recommended is Galois Theory (3rd edition) by Ian Stewart, not only is it a beautiful story but beautiful mathematics, you can "Look Inside" a large part of it on the Amazon website before you buy.
Something that is the upper end of undergraduate, but worth mentioning, is the free-online book Algebraic Topology by Allen Hatcher. Many tricky concepts, it's essentially an encyclopedia of the subject, and should be on everyone's bookshelf/stored on their computer.
A: I applaud you for taking the initiative to study on your own. I noticed the post was more than 6 months ago, so not sure if you are still looking for help. I recently started a website, www.redhoop.org, to help self learners like yourself. Search for math courses and see if you find any of them useful to you. Feel free to tell me what you think. Again, keep up your learning! Knowledge is the most important and only long lasting asset we have.
A: Hooray for mentioning Tom Korner; he has a draft text on undergraduate analysis that (if I recall correctly) makes use of Bolzano-Weierstrass and Interval Bisection but 'not too much else'. His stuff is written in a light-hearted style, with humorous footnotes and so on. If it's good enough for the students at Cambridge, then it must be worth a look. 
Also, have a look at Prof Gowers' blog and perhaps say hello. The more you interact with other learners, the more relaxed and informed you will be about discussing your studies- a good thing if you are going to interview to get onto a degree course at some point. Do you have a university near you? If so, what are the rules about visiting their maths faculty library? 
See if you can arrange to have reference rights there. in the UK, all you need is a local library card, and most university libraries will let you have a browse whenever you like.
You might feel a bit awkward at your age but I say take the plunge and go for it. Today's awkward kid (why is he here?) is tomorrow's undergrad with a place on the course.  
On a similarly confident note, have you heard of an examination called STEP? Google the word. STEP is administered by Cambridge University and extends on the current UK pre-university mathematics requirements. You could download some past papers and have a go.
If it's all too easy for you, I've just spoken to the next Ramanujan. Meanwhile, most people 
who enter for the STEP exam want to get into Cambridge to study maths and find the papers pretty difficult but a challenge worth taking on, given the rewards on offer. Perhaps you are at that age where, with a bit of luck here and there, you could make a real go of that.        
At the very least, if you download the past papers and keep a record of your work, you will have something interesting to show tutors when you apply for university.
To conclude, good luck and when you eventually get your office at the Institute of Advanced Study, tell them I sent you. Nobody will know what the heck you're on about, or who I am, but they'll laugh anyways because, hey, there goes that kid who proved that thing ! :)     
A: I am not a mathematician, but when I was learning mathematics at the undergraduate level, a great professor recommended A Radical Approach to Real Analysis. I found it informative, full of interesting problems, and quite enlightening (it couches real analysis in the problems it was developed to solve, and as such added rich context to a subject I found otherwise difficult to access).
A: I would suggest some mathematical modeling or other practical application of mathematics. Also Finite automata and graph-theory is interesting as it is further away from "pure math" as I see it, it has given me another perspective of math.
A: Here is the most rigorous/best undergraduate course i can think of:
1-First, it is best to take advantage of the four introductory courses offered by the MIT OCW. They will give you a nice background for the following course.
2- If you only know basic non rigorous calculus (like the course offered on the OCW), then start with "A Course of Pure Mathematics" by G.H.Hardy or "Calculus" by Apostl.
3- Study "Advanced Calculus" by Loomis & Sternberg, along with "Basic Algebra" by Knapp or "Algebra" by Artin.
For complex analysis, check out "Complex Analysis" by Elias Stein, or the one by Ahlfors.
4- By now, you should be ready for advanced/graduate level mathematics.It is recommended that you first study a book like "Topology" by Munkres in order to increase your knowledge of this particular field.
If you can finish the first two steps before graduating high school, that would be impressive. The books listed in (3) are typically the ones given to really advanced freshman undergrads, like Harvard's math 55 and so on.. This list will prepare you for advanced mathematics, but there is still many subjects that are not covered, like discrete mathematics and number theory.. I do not know your interests, but you should always focus on the things that interest you the most. You still have time...
