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 and 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 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 a time for a full proof but you can give a sketch using group actions on trees.
Now, 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.
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).