To start out with, I'm a junior in high school who is intrigued by the rigor of higher mathematics and is currently attempting to self study Volume 1 of Apostol's Calculus. I haven't had any previous experience with proof based mathematics, but I recently stumbled on a book that seems to suit my needs well. The book is Basic Concepts in Mathematics by Zakon, which is supposed to be read before or in conjunction with his two books on analysis. It seems to give a good exposition on set theory and additional topics in a rigorous fashion but without assuming any prior knowledge. I plan to read this alongside Apostol, but was wondering if their were any better books with regard to exposing the reader to proof based mathematics but with a focus on analysis like Zakon's text. Further, the book is free and can be accessed by the following link:
http://www.trillia.com/zakon1.html
I have gone through a small portion of the Book of Proof, but I'm looking for a text less about learning proofs than about applying them and to give me more mathematical maturity. If you have any suggestions for me or thoughts on Zakon's book, it would be greatly appreciated.
I was just recommended a few books by @JasperLoy in MSE chat. They're as follows:
Calculus
Real Analysis
I would personaly like to recommend Anton Bivens' Calculus Text to compliment Spivak. Also, Kreyzig Advanced Mathematics is also a useful stepping stone in any math journey. Try out Iridov and also Integral Kokoboken (Cookbook) if you like challenges.
If you're only starting out, I suggest going through various math textbooks like Playing With Numbers and Mathematics X/XI/XII from syllabii set by ICSE, CBSE, GCSE, iB, et cetera.
Best of Luck.