Differential Geometry past an introductory course?

You can read Introduction to Topological Manifolds, Introduction to Smooth Manifolds and Riemannian Manifolds by John Lee in that order.

I absolutely, with no hesitation, suggest, in fact, I implore you to read Lee's Riemannian Manifolds:An Introduction to Curvature. As is true of all of Lee's books, it is the clearest exposition on the majority of topics relating to the books contents. The highlights of the book though are his constant vigilance in keeping you attentive to the intimate connections of the topology of a manifold and its geometry and his unmatched (in my humble opinion of course) explanation of connections and why they're useful.

The other book that I recommend highly is Jost's Riemannian Geometry and Geometric Analysis. While this book has good exposition, it's much more of a "toolkit" book than any of the one's you've mentioned. In particular, it really does give you a great hands-on introduction to geometric analysis, a tool which will be indespensable if you decide to go further into (analytic) geometry than a first course on Riemannian manifolds