This is the natural starting point to transition from computational math to proof-based math. In my formal studies I was always intimidated by proofs and coming up with a rigorous argument always seems like a black art to me. However, studying this topic totally demystified them for me and now I very much enjoy developing and writing proofs. Each proof is reminiscent of a unique puzzle or brain-teaser to get from the premises to the conclusions.
In a lot of mathematics programs at universities, a dedicated course in this is optional and some students first take an analysis or abstract algebra course and pick up proof techniques along the way. Other universities have a course called Discrete Mathematics that teaches this along with basic set theory. In my opinion it is well worth it to study this topic on its own before attempting to learn other higher math topics.
The textbook I chose to self-study this topic is the excellent “How to Prove It” by Daniel J. Velleman. I cannot recommend this text highly enough. It requires almost no prerequisite knowledge beyond high school algebra, though calculus is helpful in a few spots as well. It starts from the very basics of logic then teaches the structure of proofs and techniques for proving various forms of logical statements. In the subsequent chapters it teaches a good amount of basic set theory, which is very useful for studying other topics, and also provides great material on which to practice writing proofs.