If you are serious about learning real proofs don't listen to anyone who recommends Stewart. No offense to them, but they simply don't have a clue what they are talking about.
Royden is a decent standard.
Rudin is also used a lot of places and there are others, but if you haven't read a real math book before (and no offense to the many people hawking mainstream math books, but books like Stewart are not real math books) it might be a tough read. Of the "standard" books for rigorous courses, Royden is generally thought of as the easiest read. MIT has some stuff up on
open courseware too. It's the calculus with theory, not the ordinary calc. Here are some
lecture notes derived from the ones I used at my alma mater for good measure.
Going it alone with a real math book for the first time is a little - er - bold. It is a foreign language. Much more so than the mainstream textbooks designed for the bulk of the student body. Then again, you might not actually be looking for a real proof book after all - despite how you worded your request. If you are really just looking for a reintroduction to calc, Stewart might do you just fine. 99% of people will certainly find it much more readable!