I'm not sure if the original question is about a one semester or year course.
If this is the first course the students have ever had in differential geometry, then I still agree with Anton that at least the first semester should be about only 2-dimensional manifolds embedded in $R^3$ and Gauss-Bonnet. The point here is that everything can be understood visually, but you learn how to deploy linear algebra and calculus to prove what seems obvious visually. The full power of differential geometry is displayed very nicely. Guillemin and Pollack provides a nice textbook to base the course on. I also like O'Neill's elementary differential geometry textbook.
I would not introduce the more abstract machinery until the second semester, and even then try to be selective about what is discussed because there is just too much. It seems best to focus on basic Riemannian geometry and what, say, sectional curvature means (this builds nicely on what was done in the first semester). It is of course important to introduce many different examples. Although the basic abstract definitions and properties of Lie groups and algebras could be introduced, I believe the focus should be on how to build interesting geometric spaces from standard matrix groups ($GL(n)$, $SL(n)$, $SO(n)$, $SU(n)$).