When it comes to self-studying proof-based math, a few classic books are highly recommended. "How to Prove It" by Daniel J. Velleman is an excellent choice for beginners. For a deeper dive, "Principles of Mathematical Analysis" by Walter Rudin is a classic for real analysis. "Introduction to the Theory of Sets" by Joseph Breuer is great for set theory. Don't forget to complement your studies with online resources, like lecture notes and problem sets from universities.
As an amateur, I have heard differing opinions on this topic, but I quite enjoyed working through the "Software Foundations" book: https://softwarefoundations.cis.upenn.edu/.
This book uses the Coq proof assistant to work through simple example proofs. If you stick with it for a while, the puzzle of finding proofs and getting instantaneous feedback from the assistant becomes quite an enjoyable process.
I have a similar goal as you. Initially I tried reading "How to Prove It: A Structured Approach" but I only got so far, mostly because I decided to prioritize upskilling and getting a new job first.
There's also a relevant hackernews post[0], and one of the suggestions there was "Proofs: A Long Form Textbooks by Jay Cummings".
[0] https://news.ycombinator.com/item?id=31800081