What would a graduate course on systolic geometry typically cover?

This is interesting. I imagine that any course would vary quite a bit depending on who taught it.

Any course should probably contain Gromov's proof of the systolic inequality for essential manifolds. Other than that, I am not sure. The course could dive into systoles on surfaces and some of the arithmetic constructions in Teichmuller theory, or it could develop harmonic maps and scalar curvature rigidity theorems, or it could take a dynamical systems approach and discuss the relationship between volume entropy and closed geodesics.

I have no idea. You should come up with a curriculum and post it here.