Text for an introductory Real Analysis course.

Stephen Abbott, Understanding Analysis
Strongly recommended to students who are ony getting to grips with abstraction in mathematics. Find a review here.

Anyone that thrusts baby Rudin - as so many departments do, sadly, in an act of either callous indifference or elitist zealotism - on beginning analysis students with no prior experience with rigor is committing an act of inhumanity against a fellow human being. Let's face it: Calculus just ain't what it used to be and Rudin is going to be a buzz-kill for any but the best students. I personally have never liked Rudin even for good students. Rudin seems more interested in showing how clever he is then actually teaching students analysis.
My recommended texts:

• For average students,who have never seen proofs before, I strongly recommend Ross' Elementary Analysis:The Theory Of Calculus.
  It's gentle, complete and walks the reader through a careful presentation of calculus containing many steps that are usually omitted or left as an exercise. It can also be used for an honors calculus course: I've had friends that have used it for that purpose with great success. Spivak is a beautiful book at roughly the same level that'll work just as well.
• More advanced, but I think well worth the effort, is Kenneth Hoffman's Analysis In Euclidean Space, which I reviewed for the MAA online a few months ago when Dover reissued it.
  It's an amazingly deep and complete text on normed linear spaces rather then metric or topological spaces and focuses on WHY things work in analysis as they do. This is the kind of book EVERYONE can learn something from and now that it's in Dover,there's no reason not to have it.
• Lastly, for honor students on their way to elite PHD programs, we now have a wonderful alternative to Rudin and I'm shocked no one's mentioned it at this thread yet: Charles Chapman Pugh's Real Mathematical Analysis, which developed out of the author's honors analysis courses at Berkeley.
  It's terse but written with crystal clarity and with hundreds of well-chosen pictures and hard exercises. Pugh has a real gift that's on display here. He knows exactly how many words it takes to clearly explain a concept-NOT ONE WORD MORE AND NOT ONE WORD LESS. I've never seen any author who does this as effectively as Pugh. The many, many pictures greatly assist him in this task: all of them serve some purpose, none are throwaways just to fill space. Even if it's just to make a joke(see the cornball pic in chapter one showing a Dedekind cut,ugh).
  

Oh, almost forgot my personal favorite: Steven Krantz's Real Analysis And Foundations. If I was ordered to teach real analysis tomorrow, this is probably the book I'd choose, supplemented with Hoffman. Krantz is one of our foremost teachers and textbook authors and he does a fantastic job here giving the student a slow build-up to Rudin-level and containing many topics not included in most courses, such as wavelets and applications to differential equations. What's most impressive about the book is how it slowly builds in difficulty. The early chapters are gentle, but as the book progresses, the presentation and exercises become steadily more sophisticated. By the last chapter, the presentation is a lot like Rudin's. I would strongly consider this text if I was trying for self study.

Anyhow, those are my picks.

Look no further than Spivak's completely amazing Calculus. I have taught analysis courses from this book many times and learned many things in the process. One example is the wonderful "peak points" proof of the Bolzano-Weierstrass theorem. The exercises are really good too.