except for polya, in my opinion the hundreds of books on how to do mathematics are mostly worthless. maybe not entirely, since i recall learning a tiny amount from browsing some good ones back in high school (by Max Black?).
but mostly one just needs to practice reading and
writing proofs. it helped me at the very beginning to have a little course in logic, from principles of mathematics, by allendoerfer and oakley. another good intro to logic and proofs is harold jacobs high school geometry book.
but these myriad books used in college courses with titles like introduction to higher math, or how to write proofs, or whatever, include some of the worst books we have to sort through as math professors.
the one i used last time, by vanstone and gilbert? wasn't as bad as most of them as i recall. i kind of liked it, but i don't recall it actually teaching you how to do proofs. it just had a nice selection of easy topics, was written decently, and was paperback hence hopefully not too exhorbitant.
I think i learned to do analysis proofs mostly from following a course on measure theory and functional analysis. The prof made it very clear how to do proof by contradiction, and use quantifiers, and negate statements, and choose epsilons and so on. But i do not think I learned any useful math in that course. I would much rather have had a more intuitive course in which i got a better feel for lebesgue measure and integration, than a formal course where i learned to manipulate epsilons.
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