Rudin's Principles of Mathematical Analysis or Apostol's Mathematical Analysis? I am in high school and have no access to a professor or anyone. I previously used Calculus Volume I by Tom Apostol and Spivak's Calculus (for the differential calculus bit).
I can choose between  Mathematical Analysis by Tom Apostol and  Principles of Mathematical Analysis by Walter Rudin, as I was gifted Rudin by a friend and ended up buying Apostol as well. I will be indebted if someone told me which one is the tougher one and which one is better for the self-learner.  I have no issues about how tough the book is, but I would like  the book that enables me to understand the subject better without being too compressed or too verbose and guides me better.
 A: The best advice I can give you is to do what I did when learning real analysis: Use them both. Apostol has a far better exposition, but his exercises are not really challenging. Rudin is the converse -- superb exercises, but dry and sometimes uninformative exposition. The 2 books really complement each other very well -- especially if you're self-learning. 
A: You can try "Curso de análise" vol1 and vol2 by Elon Lages Lima too
In this series of books ideas under a topological analysis approach and to some extent intuitive occur . It is harder to read than Apostol but less than Rudin, the problems are not easy so working on them will help to absorb the ideas of analysis.


*

*Vol 1 here

*Vol 2 here 

*Vol 3 here
A: I would recommend at least using Rudin as a supplement based on my own experience with PoMA and Real and Complex.  I did self-study out of PoMA and let me warn you that if you decide to go that route, it will be a very difficult struggle.  Rudin presents analysis in the cleanest way possible (the proofs are so slick that they often have more of the flavor algebra than analysis to me honestly) and often omits the intermediate details in his proofs.  You should be prepared to sit down with a pencil and paper and carefully verify all the steps in his arguments.  I don't want to talk about that though, since you can find that comment on any review of Rudin.
Let me tell you about Rudin problems.  You will stare at them for hours--days even--and make absolutely no progress.  You will become convinced that the statement is wrong, that the problem is beyond your tool-set, and you may even consider looking up the solution. If you stare at the problems long enough, you will eventually come up with the solution--and realize why he asked the question.  
I always find that the hardest part of learning a new field of math is learning what an interesting question looks like.  Rudin had exceptional mathematical taste, and that taste shines through both in those often-maligned slick proofs and in his choice of questions.  If you take the time to ask why each question was asked, how it fits into the bigger picture, and what in the chapter it connects to, you will learn an incredible amount about the flavor of analysis.  Really, if you want to learn how to think like a classical analyst, read Rudin.
As an aside, this may not be the case for you but I find that if a book is too well exposited, it actually detracts from my understanding.  Rudin may leave out details, but at least then it is known that you need to fill them in.  Doing this forced me to learn a lot of the basic argument techniques in analysis.  When using a book that carefully explains all the details, I find that it is a bit too easy to waive my hand at an argument and not spend time really learning it since the argument looks so clear. Admittedly that is possibly because I am, at heart, pretty lazy :)
A: I emphatically insist that you use Apostol. Rudin is not a bad book, but especially for someone who is looking for a first introduction to higher mathematics it's just too terse, and too unintuitive--also, the problems may be a bit hard. Moreover, Apostol is a fantastic expositor, he will also cover more of the things that someone first seeing analysis should see.
A: Perhaps you might want to consider an alternative that you hadn't mentioned. Here is a link to a beautifully presented copy of the lectures given by Fields Medal winner Vaughan Jones for his Real Analysis class. I found them most elegant, self-contained, and very accessible. They are available for free here:
https://sites.google.com/site/math104sp2011/lecture-notes
then at bottom is a link to pdf file, just tried it and works.
A: I can answer from England. In 1959
I read Tom as a Balliol undergrad. It was his 1st edition which started with a brilliant exposition
of the Riemann-Stieltjes Integral.
His 2nd edition was mundane. Noone
to date has picked up that he used
Finer Partitions rather than a Mesh
size.
I read Rudin. Both Real and Complex.I felt proofs were not fully rigorous. By today's standards. But Math is evolving.
I could recommend Zorich in the translation. Both volumes.See Springer as publisher. He teaches at Moscow State University. 
A seminal work is still Ahlfors Complex Analysis. He won a Fields
Medal for it.
I had caught up on Banach spaces with a publication from The Canadian Mathematical Society.
Jeffrey G Thomas.
