Michaek Spivak's "Calculus" is pretty great for learning real analysis. The other canonical text is "Principles of Mathematical Analysis" by Walter Rudin, which I've never used, but is purported to be of the same quality.
I wouldn't recommend either of those two book for self-teaching.
For whatever reason, I taught myself advance mathematics in High Schools. Spivak and Rubin were pretty inaccessible. Sure, they are rigorous and high level but that meant they didn't lend themselves to an easy read. "Real Analysis" by Royden was relatively quick to go through - and I had only the start of calculus. Royden gives a fairly simplistic and accessible development of higher mathematics starting with set theory.
I'd imagine that would be the most helpful.
(In my High School years, I went from algebra sophomore year to reading Royden and doing a community college calculus class junior year to passing a differential geometry course and the undergraduate honors seminar at UCLA).
If you're going to try to learn that stuff on your own you'll need other books to supplement it and provide examples, counter-examples, intuition, and alternative explanations.