Differential Geometry past an introductory course? I'll be doing an independent study with one of my profs in differential geometry next semester (my university did not happen to offer an intro diff. geometry course next semester like it usually does).  I'll be mainly working out of Barrett O'Neill's book but will also be checking out different perspectives by looking at do Carmo's book (and maybe Spivak's?)  I've been planning out the rest of my semesters and even if I end up taking courses in a wide range of branches in mathematics, I'll still have quite a bit of free credits to delve more deeply into one subject.  If I do choose to go further into differential geometry, what are some important classes to take/books to read?  Books I've looked into so far are Do Carmo's Riemannian Geometry, Barrett O'neill's Semi-Riemannian Geometry, as well as differential topology books like Milnor's topology from a differentiable viewpoint or Lee's introduction to smooth manifolds (I understand these are important for more advanced work in differential geometry?)  What is the recommended order I should learn these subjects in?  Any other suggestions/recommendations?  
Thanks!
 A: You can read Introduction to Topological Manifolds, Introduction to Smooth Manifolds and Riemannian Manifolds by John Lee in that order.
A: 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
A: Some time ago I have written some lecture notes on general relativity, and I started with an introduction to differential geometry. Maybe you will find it useful after you had an introductory course. Here it is (free download, FDL):  https://sites.google.com/site/winitzki/index/topics-in-general-relativity
A: I highly recommend an older book, Notes on Differential Geomtry by Hicks.  Where it really excels is how it starts completely extrinsically, and introduces all of the basic concepts such as connections for hypersurfaces, and shows how natural they are, and how they satisfy a natural set of axioms.  Then he proceeds to the more intrinsic viewpoint, with you already in possession of intuition of the point of a connection.
I would also recommend skimming Misner, Thorne, and Wheeler's Gravitation, which is differential geometry aimed at physicists for the purposes of teaching general relativity.  It's a good source for intuition from a different point of view.
A: When you are trying to get a deeper insight into the subject, I would also recommend the book of Jeffrey M. Lee: Manifolds and Differential Geometry.
A: I would recommend the book "Lectures on Differential Geometry" by Shlomo Sternberg.
Covers quite fundamental topics such as De Rham's theorem and the Generalised stokes theorem.
The book explores various other sub-topics not usually covered in a typical introductory course.
