Talking about finitely generated, residually finite groups the first result that comes to my mind (apart the well-known fact that every free-group is residually finite) is the following powerful result:
Theorem (Malcev, 1940) Every finitely-generated linear group is residually finite.
See
A. I. Malcev: On the faithful representation of infinite groups by matrices, Mat. Sb. 8 (50) (1940), 405-422; English transl., Amer. Math. Soc. Transl. (2) 45 (1965), 1-18. MR 2, 216.
Other interesting examples are of finitely generated, residually finite groups are:
- the one-relator group $$\langle a, \, t\; | \; a^{t^2}=a^2 \rangle,$$ which is residually finite but not linear (Drutu-Sapir);
- fundamental groups of type $\pi_1(S)$, where $S$ is a compact surface;
- Baumlag-Solitar groups of type $B(m, \, m)$, $B(n, \, 1)$ and $B(1, \, m)$, where $$B(m, \, n) = \langle a, \, b \; | \; a^{-1}b^ma=b^n\rangle.$$
See this blog post on the work on Bausmlag, that also contain many other interesting results, mostly about the relationships between residually finite and Hopfian groups.