(I had tried to add a clarifying comment to @AlexanderWoo's good answer, but it was mysteriously deleted.)
My point was, and is, that it is not constructive to think of "proofs" as a thing separate from normal human discussion of things. Rather, proofs are really explanations, or discussions that persuade. There is no magic formula, and "proof" is not an alien thing.
Yes, many math curricula (perhaps especially in the U.S., these days) in high school and the beginning of college or university have essentially no content that is about explanation or persuasion, but only about inexplicable rules.
I do have some acquaintance with various "introduction to proof" textbooks and courses, and I am not a fan of any of them, for various reasons. Instead, as suggested in Alex W's answer, simply reading relatively easy proofs of real things (rather than contrived) is the way to learn what the point of "proofs" is. It's not conforming to rules in a ritual, it's about careful appraisal of causality and such...
One particular problem in terms of understanding mathematics, is the notion that we should not trust our physical intuition. It is perverse to invalidate beginners' self-confidence and critical judgement by telling them things like this. Mathematics is not about inscrutable stuff behaving wildly contrary to our physical experience, and it is perverse to portray it as such. One iconic example is the intermediate value theorem in calculus. On one hand, it is "obviously true"... but it is non-trivial to prove, because a precise notion of "real number" is necessary, to begin. But that should not make us doubt the fact.
Especially when I hear people say "what should I memorize?" it worries me. Of course, memorization can be a first step toward understanding, but I strongly prefer thinking of "talking about math" as more akin to throwing and catching a ball, or walking across a room. Maybe it can be axiomatized (and that may be of some interest), but it seems that our best understanding of it is achieved by viewing it as "real", as opposed to disconnected from our reality.
So, sure, look at "intro to proofs" books, if they are fun, and do exercises in them, if they're fun, but that's further from real math than "T-ball" (hitting a ball off a "tee", for little kids) is from baseball.
Better to look at relatively elementary texts that address some bit of mathematics itself, rather than claiming to talk about "proofs".