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.