In addition to the dihedral groups, the quaternion group $Q_8$, the symmetric and alternating groups, and small matrix groups, I have always found the following examples useful:
Automorphism groups of graphs: I usually introduce these at around the same time I introduce permutations. This gives a wealth of interesting examples of groups, and also gives a simple example of how groups can be used to understand other mathematical structures. There are lots of small graphs (e.g. trees) that provide simple examples of quotients and normal subgroups, and you can also use graphs to give nontrivial applications of various group action concepts. For example, I have a homework problem where students compute the order of the automorphism group of the Petersen graph using the orbit-stabilizer theorem.
Unit groups: Gallian's book places an emphasis on the unit groups $U(n)$, i.e. the group of units in the integers modulo $n$. I really like the emphasis on these groups, because their structure is not obvious, so students really have to dig into them to figure them out. For example, listing the subgroups of $U(15)$ or $U(24)$ is an interesting exercise, as is computing the automorphisms of either of these.
Symmetry groups of polyhedra: Others have already mentioned these so I won't go into too much detail here, but I should point out that lots of polyhedra beyond the platonic solids and prisms have interesting symmetry groups. For example, one of my homework problems asks the students to compute the symmetry group of the tetrahedron with vertices at $(8,8,1)$, $(8,-8,-1)$, $(-8,8,-1)$, and $(-8,-8,1)$. I'm also fond of antiprisms.