There was a time when I avoided math proofs, but now I am starting to enjoy them. I am taking Intro to Linear Algebra and am falling in love with proofs. Are there any introduction to mathematical proofs books that blow the others out of the water?
Introductory Books – Best Books for Learning Mathematical Proofs
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George Polya's How to Solve It immediately comes to mind. I know many now fantastic pre-mathematicians who learned calculus and the basics of analysis from Spivak's Calculus and even if you know the material to go back and do it again in a formal way is very healthy. In addition Proofs from THE BOOK was mentioned above and was recommended to me by Ngo Bao Chao when I asked about books to study problem-solving techniques from. I don't mean to come off as name-dropping but I feel that (as he is a fields medalist) his advice is worth heeding. I, personally, really liked it.
However I have to make note that I think if you'd phrased your question as "should I read a book about proofs to learn proofs" my response would be an emphatic no. In my experience if you don't see proofs by doing some fun mathematics you will not get much better about doing them yourself. Just reading about how to prove things can only get you so far before you're sort of stumped as to how to proceed. I would say the better approach is to find a rigorous treatment of a subject that you're very interested in, and read that, following along with the proofs of the theorems in the book and eventually trying to do them yourself without looking at the proofs given.