Geometric group theory is a huge subject, and a course that really tried to cover all of it would be too disjointed to be useful. If I were teaching such a course, I would choose a few major theorems and spend time developing all the tools you would need to prove and appreciate them. For instance, I think you could easily spend a semester focusing on:
Mostow rigidity, perhaps proved using Gromov's approach using the Gromov norm (which is probably more useful to the typical GGT student than the analytical tools needed for the original proof).
Gromov's theorem on groups of polynomial growth, which would also give you a nice opportunity to talk about large-scale geometry in general.
Stallings's theorem on ends of groups, together with some applications (my favorite would be the fact that groups of cohomological dimension one are free, but that might be too algebro-topological for you). This would require also developing Bass-Serre theory.
Three other topics that would make sense to include are hyperbolic groups, CAT(0) geometry, and the theory of cube complexes. But that would almost be a separate course, and would not have much overlap with the three big theorems above.