I'm trying to teach myself Calculus with Tom Apostol's first volume, and I love it but I am having trouble with all the exercises which require proofs. The only time I've done proofs myself was in high school geometry, and while I can follow complicated proofs by others I am confused as to how one "proves" some of the very basic statements arising from the field axioms. Is there a book or website which would prepare me for this kind of thing, or some general principles to guide me?

In addition I am a little confused as to what structure the proofs should take. Are they just a series of numbered statements? How much non-mathematical language can you use? Thanks!