Here's a few examples in line with classical combinatorial group theory.

1. Though small cancellation groups as a whole have already been mentioned, one important subclass of these are the one-relator groups with torsion $\langle A \mid w^n = 1 \rangle$, which were shown to be linear rather recently. Baumslag conjectured in the 60s that they would be residually finite, and Allenby-Tang resolved a good number of special cases of this quite some time before the recent work of Agol-Wise-Haglund settled it completely.
2. Polycyclic groups (due to Hirsch) and f.g. metanilpotent groups (due to P. Hall).
3. Not all one-relator groups are residually finite, as the example non-Hopfian Baumslag-Solitar group $\langle a, b \mid b^{-1} a^2 b = a^3 \rangle$ shows. However, an important class is the class of cyclically pinched one-relator groups, i.e. those which admit a presentation of the form $\langle A \cup B \mid U = V\rangle$, where $U$ is a word over the generators $A$ and their inverses, and $V$ is one over $B$ and their inverses (and $A \cap B = \varnothing$). Baumslag, again, showed that these are residually finite in 1969. Some familiar examples, which have already appeared in other answers, are the fundamental groups of compact surfaces.
4. Baumslag showed in 1963, in an exceptionally short and beautiful paper (the proof is one paragraph long, and the remainder of the paper is a couple of paragraphs of applications) that the automorphism group of a finitely generated residually finite group is again residually finite. In particular, the automorphism group of the automorphism group of a free group is residually finite -- I do not know whether there is any other proof of this fact available using only the known presentations for these groups.
5. Finally, a non-example, but perhaps of some relevance/interest: it was once conjectured (by G. Lallement in 1974, in a paper on semigroup theory!) that positive one-relator groups are residually finite. Here a positive one-relator group is one $\langle A \mid w = 1 \rangle$ where $w \in A^\ast$ is a word not including any inverse symbols. However, as noted by Perrin-Schupp, the non-Hopfian Baumslag-Solitar group $BS(2, -3)$ admits the positive presentation $\langle a, b \mid (ab)^2(ba)^3 = 1 \rangle$.

Here's a few examples in line with classical combinatorial group theory.

1. Though small cancellation groups as a whole have already been mentioned, one important subclass of these are the one-relator groups with torsion $\langle A \mid w^n = 1 \rangle$, which were shown to be linear rather recently. Baumslag conjectured in the 60s that they would be residually finite, and Allenby-Tang resolved a good number of special cases of this quite some time before the recent work of Agol-Wise-Haglund settled it completely.
2. Polycyclic groups (due to Hirsch) and f.g. abelian-by-nilpotent groups (due to P. Hall).
3. Not all one-relator groups are residually finite, as the example non-Hopfian Baumslag-Solitar group $\langle a, b \mid b^{-1} a^2 b = a^3 \rangle$ shows. However, an important class is the class of cyclically pinched one-relator groups, i.e. those which admit a presentation of the form $\langle A \cup B \mid U = V\rangle$, where $U$ is a word over the generators $A$ and their inverses, and $V$ is one over $B$ and their inverses (and $A \cap B = \varnothing$). Baumslag, again, showed that these are residually finite in 1969. Some familiar examples, which have already appeared in other answers, are the fundamental groups of compact surfaces.
4. Baumslag showed in 1963, in an exceptionally short and beautiful paper (the proof is one paragraph long, and the remainder of the paper is a couple of paragraphs of applications) that the automorphism group of a finitely generated residually finite group is again residually finite. In particular, the automorphism group of the automorphism group of a free group is residually finite -- I do not know whether there is any other proof of this fact available using only the known presentations for these groups.
5. Finally, a non-example, but perhaps of some relevance/interest: it was once conjectured (by G. Lallement in 1974, in a paper on semigroup theory!) that positive one-relator groups are residually finite. Here a positive one-relator group is one $\langle A \mid w = 1 \rangle$ where $w \in A^\ast$ is a word not including any inverse symbols. However, as noted by Perrin-Schupp, the non-Hopfian Baumslag-Solitar group $BS(2, -3)$ admits the positive presentation $\langle a, b \mid (ab)^2(ba)^3 = 1 \rangle$.