I had a very short introduction to number theory in one of the classes I took and I learned a bit about divisibility and congruence, not much further than Fermat's little theorem, and I would like to learn more.
What's would be a good book that covers the basics and takes on a good level of depth, possibly with utmost rigorous proofs and a good variety of examples?
After that, what would be a bit more advanced titles?
Silverman's A Friendly Introduction to Number Theory is a good text to start with, although it is very computational, but does go through basic ideas such as modular arithmetic and some basic diophantine equations.
Alternatively, Tattersall's Elementary Number Theory in Nine Chapters is available online: http://www.fmf.uni-lj.si/~lavric/Tattersall%20-%20Elementary%20number%20theory%20in%20nine%20chapters.pdf. This doesn't cover quite as much stuff (for example quadratic reciprocity) but does go a bit more in depth and has plenty of proofs with it.
After that, if you like the algebra side of things, elliptic curves are done well in Silverman-Tate's Rational Points on Elliptic Curves but you will need to have a reasonable understanding of abstract algebra for that (things like groups, fields etc). Elliptic curves are equations of the form $y^2=x^3+Ax+B$ and have a strong connection to Fermat's Last Theorem.
In terms of a comprehensive introduction, Elementary Number Theory by Underwood Dudley is a nice text with plenty of exercises and thorough proofs as far as I can tell. I am currently working through it and enjoy it very much. I can't speak much for more advanced texts, however.
