Here is my take. Unlike Andy, I would not structure such a course around big theorems. In part, this is because your students simply do not have enough background to handle any "big theorems." Instead, I would try to emphasize "small things" and connections of geometric group theory with various areas of mathematics. Item 2 is not, strictly speaking, geometric, but, IMHO, belongs to any geometric group theory course.
Free groups and group presentations (likely covered in a basic algebraic topology class). Dehn problems in combinatorial group theory and undecidability results (without proofs).
Residual finiteness in general; residual finiteness of finitely generated subgroups of $SL(n,\mathbb Z)$ and statement (without a proof) for finitely generated general matrix groups. Application to decidability of the WP. Mikhailova example of undecidability of the membership problem for subgroups of $SL(4,\mathbb Z)$.
General mantra of "groups as geometric objects": Cayley graphs, Cayley complexes, quasi-isometries, MS Lemma. Surface groups and hyperbolic plane, abelian groups and Euclidean spaces.
Generalities of hyperbolic groups, isoperimetric inequalities, Dehn algorithm and decidability of the WP. Morse Lemma and quasi-isometry invariance of hyperbolicity.
Basics of small cancellation theory and why/when small cancellation implies hyperbolicity.
Quasiconvex subgroups and decidability of the membership problem.
Group actions on simplicial trees, amalgams of groups and relation to the Seifert - Van Kampen Theorem.
Ends of spaces as topological invariants and ends of groups as coarse geometric invariants. Statement of the Stallings Theorem: It is unlikely that you will have time for a full proof but you can give a sketch using group actions on trees.
Now (maybe even earlier!), you probably are out of time. If not:
Probabilistic aspects (assuming that your students took a basic discrete probability class). Gromov's density model and at least a statement of hyperbolicity of random groups with a sketch of a proof using small cancellation theory. This would provide a rather satisfactory conclusion: In the beginning (item 1) they learn that "nothing can be done" but at the end of the course (item 9), they learn that "everything can be done asymptotically almost surely." Just as in Hegelian "Logic."
Instead of 9: Statement of the Mostow Rigidity Theorem and an outline of a proof using "zooming in" argument, blackboxing the required analytical details (do not even try to explain what quasiconformal maps are, just state the needed properties).