Are you looking for a complete classification? It seems unlikely that it can be achieved.

At any rate, 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.

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.