Rudin's Principles of Mathematical Analysis or Apostol's Mathematical Analysis?

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.

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 :)

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.