My take on it is like this:
Smooth manifolds
- Loring Tu, Introduction to manifolds - elementary introduction,
- Jeffrey Lee, Manifolds and Differential geometry, chapters 1-11 - cover the basics (tangent bundle, immersions/submersions, Lie group basics, vector bundles, differential forms, Frobenius theorem) at a relatively slow pace and very deep level.
- Will Merry, Differential Geometry - beautifully written notes (with problems sheets!), where lectures 1-27 cover pretty much the same stuff as the above book of Jeffrey Lee
Basic notions of differential geometry
Jeffrey Lee, Manifolds and Differential geometry, chapters 12 and 13 - center around the notions of metric and connection.
Will Merry, Differential Geometry - lectures 28-53 also center around metrics and connections, but the notion of parallel transport is worked out much more thoroughly than in Jeffrey Lee's book.
Sundararaman Ramanan, Global calculus - a high-brow exposition of basic notions in differential geometry. A unifying topic is that of differential operators (done in a coordinate-free way!) and their symbols.
What I find most valuable about these books is that they try to avoid using indices and local coordinates for developing the theory as much as possible, and only use them for concrete computations with examples.
However, the above books only lay out the general notions and do not develop any deep theorems about the geometry of a manifold you may wish to study. At this point the tree of differential geometry branches out into various topics like Riemannian geometry, symplectic geometry, complex differential geometry, index theory, etc., not to mention a close sister, differential topology.
I will only mention one book here for the breadth of topics discussed
- Arthur Besse, Einstein manifolds - reviews Riemannian geometry and tells about (more or less) the state of the art at 1980 of the differential geometry of Kähler and Einstein manifolds.