Self-teaching mathematics I graduated from high school 5 years ago, decided not to go to university, so life has taken me in a different direction. However, ever since I graduated I've been half-seriously interested in mathematics, learning things that seemed interesting to me (the interests were all over the place, so my knowledge of math is patchy). 
Now, I have the desire to learn math on my own in a more or less linear and formal fashion. My goal is to be somewhat competent in the "basic" branches of mathematics taught in university.
My question: (1) What are the basic branches of mathematics taught in university that I absolutely must learn (don't hesitate to give a relatively long list)? (2) in what order should I learn them, and (3) what are the best textbooks for learning them on your own?
 A: My advice would be to do an Open University maths degree or just buy their course materials from e-bay. That way you would get a broad maths education with really well explained materials. You would also learn how to read maths text books for that is a skill in itself. If you are not based in the UK just have a look on-line and get the access course materials on e-bay to see what you think. The nice thing is that they start right from the beginning assuming no real maths knowledge.
A: A good list of topics and as a bonus a list of free online sources can be found on the American Institute of Mathematics Open Textbook Initiative: https://aimath.org/textbooks/approved-textbooks/
Now as for order, you could start at the first column on the left, then move down and then move on to the second column, etc...  It is not a perfect ordering and you can skip and jump around (most likely the very first few you've seen already).  But more or less this is a good representation of topics one would encounter in an undergraduate math degree, and for each topic a couple of texts that have been used in such classes and can be used for free online.
As for how to self-learn.  It is very similar to learning in class, but requires way more self-motivation (there is no threat of a bad grade).  Best is to go section by section, and after each section try to do as much of the exercises as possible.  There is really no replacement for doing exercises.  Do not worry if you struggle with exercises.  If they are too easy, then you are not learning much.  As they say:  No pain ... no gain.
Many textbooks usually have somewhere in the introduction at least a hint to the lecturer on the best possible course.  If you do self study ... you are the "lecturer".
In the end, I would say the difference in textbooks is not that great.  There exist terrible textbooks out there to be sure, but any textbook that has been used in a couple of classrooms is most likely good enough for self study.  What will make the most difference is how you yourself approach learning, and how much motivation you have.  Best might be to fix a schedule for yourself: set aside time for reading, a schedule on how much you want to get through each time, then set aside time for doing exercises for the sections you've read.
A: As a computer science undergrad, I recommend linear algebra and discrete math. Discrete math really makes you think and come up with effective approaches to solve a problem. In terms of books, visit your college and university library, and they usually have the books taught at the school in the reserve section. Find which book is necessary for what courses. You can also find and download the PDF version of the books online.
Here are my suggested books:


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*Contemporary Linear Algebra by Howard Anton and Robert Busby

*Discrete and Combinatorial Mathematics (5th Edition) by Ralph P. Grimaldi


Here is the website for the courses and materials necessary for getting a Math degree at Simon Fraser University. By clicking on each course, you can see the list of topics that are covered, and it will tell you the books that are being used for the course.
http://www.sfu.ca/students/calendar/2018/fall/programs/mathematics/major/bachelor-of-science.html
http://www.sfu.ca/students/calendar/2018/fall/programs/applied-mathematics/major/bachelor-of-science.html
A: I have only ever been educated in the U.K. so I will reference how things work there in my answer.
Here are a list of “must haves,” which is really a list of “things a mathematical non-mathematician can be expected to know about; things a mathematician can be expected to know about (although maybe not in detail or al of these); and things that form the basis of something a mathematician might study”


*

*(Most of these should be prefixed by “elementary”) (a) Further maths catchup and extension (a mix of calculus/differential equations, basic “linear algebra,” ie vectors and matrices from an applied-maths viewpoint. These should extend to some classical ode solving techniques, multivariable linear first order systems of odes (mixed with the linear algebra), phase planes, Cauchy-Schwartz, eigenthings, some “special” matrices (symmetric,unitary,orthogonal,hermitian). (b) Some introduction into “how to do proofs and some foundational objects of mathematics,” a non-formal introduction to logic, integers, sets. Some basic number theory (eg up to Chinese remainder theorem and FTA), an understanding of what it means to be a set, function, relation, equivalence relation, injection, subjection, bijection (but without eg ZFC), the weak/strong principle of induction. What a rational or real number is. This should set up enough to cope with analysis. (c) group theory. I think this should start at the definition of a group (shouldn’t really require much of a definition of a set but could many examples will be founded in number theory), then work up through homo/isomorphism, kernel/image to various basic theorems (Cayley, Cauchy, Lagrange) and up to actions (and orbit stabiliser) (d) probability, ie what is a probability space, what is expectation, what is conditional probability. Also indicator functions and a handful of distributions. Laws of large numbers. (e) Some physics. I think some classical dynamics and special relativity is good. Some people don’t think this really counts as maths but I think one should include applied maths/physics as well as the pure. American courses tend to have a bit more of a pure bent. (f) Real analysis. This should start at sequences and series (maybe include radius of convergence), move on to continuity then differentiation, then Riemann integration. Finish with some fundamental theorems of calculus. (g) More calculus. This time vector calculus. This shouldn’t be very focused on proofs and more on applications (esp to physics). Include mv derivatives and crucially suffix notation and the idea of a tensor. (h) Topology: definition of a metric space, open set, continuous function, homeomorphism, closed set, product space, quotient space, definition of a topology, some strange topologies, don’t go so far as homotopy.

*(a) Linear algebra. A bunch of definitions. A bunch of proofs. Not super fun but relevant to later maths. (b) Real analysis for multiple variables. (c) Statistics. An extension of probability to things like likelihood and estimators and such. (d) Some general applied maths techniques, Sturm-Liouville theory, Fourier series, Fourier transform, Laplace transform. (e) Some complex calculus or analysis. Things like Laurent series, poles and contour integration. (f) More algebra. Other isomorphism theorems for groups, categorisation of abelian groups, (commutative) rings, maybe eg fields or modules. Maybe some other introductions: electromagnetism, fluid dynamics, quantum mechanics, variational principles for physics calculations, some non-algebraic geometry, graph theory

*Other things to do as per interest: QM, general relativity, algebraic topology, algebraic geometry, Galois theory, representation theory, more number theory, towards algebraic number theory, functional analysis, more fluids, more formalised probability, more advanced analysis (eg lebesgue integration), formal logic
For order I would say roughly in 1. 2. 3., following interests. Some things can be done out of order. Eg graph theory and number theory have basically no prerequisites apart from mathematical maturity. As for books, I think the best books tend to be introductory texts targeted at students but these are hard to begin without any mathematical background. I don’t think recommendations are very helpful as I’ve only ever read about one book per course and that’s true of most people so I’m not really in a situation to compare books.
