TITLE: Teaching myself differential topology and differential geometry
QUESTION [208 upvotes]: I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.
I have decided to fix this lacuna once for all. Unfortunately I cannot attend a course right now. I must teach myself all the stuff by reading books.
Towards this purpose I want to know what are the most important basic theorems in differential geometry and differential topology. For a start, for differential topology, I think I must read Stokes' theorem and de Rham theorem with complete proofs. 
Differential geometry is a bit more difficult. What is a connection? Which notion should I use? I want to know about parallel transport and holonomy. What are the most important and basic theorems here? Are there concise books which can teach me the stuff faster than the voluminous Spivak books?
Also finally I want to read into some algebraic geometry and Hodge/Kähler stuff.
Suggestions about important theorems and concepts to learn, and book references, will be most helpful.

REPLY [5 votes]: My take on it is like this:
Basics of 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.
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. 

Differential topology

Amiya Mukherjee, Differential Topology - first five chapters overlap a bit with the above titles, but chapter 6-10 discuss differential topology proper - transversality, intersection, theory, jets, Morse theory, culminating in h-cobordism theorem.
Raoul Bott and Loring Tu,  Differential Forms in
Algebraic Topology - a famous classic; not a book on differential topology - as the title suggests, this is a treatment of algebraic topology of manifolds using analytic methods