As a current Harvard math grad student I think you should read many different easy books to learn a subject whenever possible, especially if you can find them for free. When you say you are mathematically able it is unclear what level you are at. All of my favorite books for learning involve huge number of exercises, and I recommend you do all of them instead of reading ahead.

For basic real analysis, my favorite book is Rosenlicht's Introduction to Analysis but baby Rudin is pretty good too, and I recommend you flip back and forth between them both.

For learning math in general, I think real analysis is a poor place to start, but that may be personal preference because I have a more algebraic slant. I highly recommend books like Herstein's Abstract Algebra, Mathematical Circles: A Russian Experience, I.M. Gelfand's Trigonometry, and Robert Ash's Abstract Algebra: The Basic Graduate Year, mostly for the wealth of exercises. Some of these are books for small children and I think those are the best sort of books to first learn from.