Thierry Aubin's book "A course in differential geometry" is really good for an introductory course. It covers the basic definitions of manifolds and vector bundles, orientability and integration (Stokes formula) and then focuses on Riemannian geometry defining the Levi-Civita connection, curvature tensor etc...
The only important missing topics are Lie groups and de Rham cohomology. Many courses in differential geometry don't talk about these subjects leaving them to specialised courses in Lie theory or Algebraic topology but I think it's a mistake.
