I'm currently taking an introductory course in real analysis at the University of Glasgow. The set text is "Calculus" by Spivak. Totally deserving of its reputation. It's a great read with loads of exercises of varying degrees of difficulty. I also dip into a few others on a regular basis:
- "Calculus", Vols. 1 and 2 by Apostol - a bit drier than Spivak but the exposition is spot on. Great coverage of topics in linear algebra too.
- "A First Course in Mathematical Analysis" by Burkhill - an oldie but a goldie. Surprised it hasn't been mentioned yet.
- "Introduction to Real Analysis" by Bartle and Sherbert - formal, well laid out.
- "Fundamentals of Mathematical Analysis" by Haggarty - a bit more hand holding. A great first text for self study I would say.