Basic book about mathematical proofs
There are books devoted to your question, e.g., Daniel Solow's book, How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000562.html but I'm not sure that's the way to go. It might be better to just pick up a good textbook in Number Theory (like Niven, Zuckerman, and Montgomery) or Abstract Algebra (like Fraleigh) or Discrete Math (maybe Grimaldi, or Brualdi), to see how people actually do proofs when they need them.
I just came across this physics forum with a thread on how to write mathematical proofs. At first I was going to copy the list of texts there here, sorting out possible duplicates. But there are so many resources listed: books, tutorials, university-sponsored guides, etc. that I'll simply steer you to the site.
Read Pólya's How to Solve It. See also these slides.
You might also like this: How to Solve it: Modern Heuristics. I've read the first few pages and it's not bad at all.
There are many mathematics texts, ranging from the middle school level to the undergraduate level, that are designed, at least in part, to serve as an introduction to proof. I would recommend that you select a text of this nature about a mathematical or allied field that you find interesting.
Three examples, off the top of my head:
- Tom Apostol: Calculus, Volume I
- James Munkres: Topology
- Uhh … I can't find it right now, but there's an introductory real analysis text called something like Analysis that I used as a college freshman which is designed that way.
If you're into computer science, you could pick up a lot of proof techniques (and exercises) from Donald Knuth's The Art of Computer Programming, but that might be a bit on the intimidating side.
My introduction to proof came from my seventh grade geometry teacher, Darlyn Counihan, whose homework and tests consisted of nothing but proofs. Sadly, I hear that most geometry classes in the U.S. these days don't require students to write a single proof, which makes me wonder just what exactly such classes are for.