The theorems in this list are in many cases central topics in 1+ semester courses, and aren’t really worth worrying about out of context. Overall you are looking at more than 5 years of full-time (40 hours/week) study, maybe more than 10 years. Unless you have extensive background reading and writing proofs, self-study is not likely to be very effective. You could conceivably make your way through with textbooks and the help of an expert tutor/mentor.

For most people the easiest method would be to enroll in an undergraduate mathematics degree at a decent university.

If you wanted to make your way through more of the topics in this list than an undergraduate degree covered, you could follow-up by enrolling in a PhD program.

There are standard texts on the various topics listed in the contents. There are online texts on these topics as well as free to download texts. To learn everything, get a math degree or self-study the curriculum of one. The prerequisite is basically calculus.

The thing about this text is it seems like a more or less random list of theorems - it lists one theory, I think, Zorn's lemma, under topology where Zorn's lemma is more a set-theory-link-to-topology (one of the guises of the Axiom of choice, really). Which to say it has next-to-nothing on point-set topology proper. Other theory listings seem just as random. So I couldn't what order you'd follow trying to learn everything specifically here.