Teaching myself differential topology and differential geometry 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.
 A: I would like to recommend Modern Differential Geometry of curves and surfaces with Mathematica, by Alfred Gray, Elsa Abbena, and Simon Salamon.  You can look at it on Google books to decide if it fits your style.  If you are a Mathematica user, I think this is a wonderful avenue for self-study, for you can see and manipulate all the central constructions yourself.  I use Gray's code frequently; I was a fan.
PS. Here is how he died: "of a heart attack which occurred while working with students in a computer lab at 4 a.m."!
A: 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
A: I would recommend Jost's book "Riemannian geometry and geometric analysis" as well as Sharpe's "Differential geometry".
The first book is pragmatically written and guides the reader to a lot of interesting stuff, like Hodge's theorem, Morse homology and harmonic maps.
The second book is mainly concerned with Cartan connection, but before that it has an excellent chapter on differential topology. Furthermore it treats Ehresmann connections in appendix A.
A: For differential topology, I would add Poincare duality to something you may want to know. A good textbook is Madsen and Tornehave's From Calculus to Cohomology. Another nice book is John Lee's Introduction to Smooth Manifolds. 
For differential geometry, I don't really know any good texts. Besides the standard Spivak, the other canonical choice would be Kobayashi-Nomizu's Foundations of Differential Geometry, which is by no means easy going. There is a new book by Jeffrey Lee called Manifolds and Differential Geometry in the AMS Graduate Studies series. I have not looked at it personally in depth, but it has some decent reviews. It covers a large swath of the differential topology, and also the basic theory of connections. (As a side remark, if you like doing computations, Kobayashi's original paper "Theory of connections" is not very hard to read, and may be a good starting place before you jump into some of the more special-topic/advanced texts like Kolar, Slovak, and Michor's Natural operations in differential geometry.)
A book I've enjoyed and found useful (though not so much as a textbook) is Morita's Geometry of differential forms. 
I can't help you with algebraic geometry. 
A: ADDITION: I have compiled what I think is a definitive collection of listmanias at Amazon for a best selection of books an references, mostly in increasing order of difficulty, in almost any branch of geometry and topology. In particular the books I recommend below for differential topology and differential geometry; I hope to fill in commentaries for each title as I have the time in the future.
If you want to have an overall knowledge Physics-flavored the best books are Nakahara's "Geometry, Topology and Physics" and above all: Frankel's "The Geometry of Physics" (great book, but sometimes his notation can bug you a lot compared to standards).
If you want to learn Differential Topology study these in this order: Milnor's "Topology from a Differentiable Viewpoint", Jänich/Bröcker's "Introduction to Differential Topology" and Madsen's "From Calculus to Cohomology". Although it is always nice to have a working knowledge of general point set topology which you can quickly learn from Jänich's "Topology" and more rigorously with Runde's "A Taste of Topology".
To start Algebraic Topology these two are of great help: Croom's "Basic Concepts of Algebraic Topology" and Sato/Hudson "Algebraic Topology an intuitive approach". Graduate level standard references are Hatcher's "Algebraic Topology" and Bredon's "Topology and Geometry", tom Dieck's "Algebraic Topology" along with Bott/Tu "Differential Forms in Algebraic Topology."
To really understand the classic and intuitive motivations for modern differential geometry you should master curves and surfaces from books like Toponogov's "Differential Geometry of Curves and Surfaces" and make the transition with Kühnel's "Differential Geometry - Curves, Surfaces, Manifolds". Other nice classic texts are Kreyszig "Differential Geometry" and Struik's "Lectures on Classical Differential Geometry".
For modern differential geometry I cannot stress enough to study carefully the books of Jeffrey M. Lee "Manifolds and Differential Geometry" and Liviu Nicolaescu's "Geometry of Manifolds". Both are deep, readable, thorough and cover a lot of topics with a very modern style and notation. In particular, Nicolaescu's is my favorite. For Riemannian Geometry I would recommend Jost's "Riemannian Geometry and Geometric Analysis" and Petersen's "Riemannian Geometry". A nice introduction for Symplectic Geometry is Cannas da Silva "Lectures on Symplectic Geometry" or Berndt's "An Introduction to Symplectic Geometry". If you need some Lie groups and algebras the book by Kirilov "An Introduction to Lie Groops and Lie Algebras" is nice; for applications to geometry the best is Helgason's "Differential Geometry - Lie Groups and Symmetric Spaces".
FOR TONS OF SOLVED PROBLEMS ON DIFFERENTIAL GEOMETRY the best book by far is the recent volume by Gadea/Muñoz - "Analysis and Algebra on Differentiable Manifolds: a workbook for students and teachers". From manifolds to riemannian geometry and bundles, along with amazing summary appendices for theory review and tables of useful formulas.
EDIT (ADDED): However, I would argue that one of the best introductions to manifolds is the old soviet book published by MIR, Mishchenko/Fomenko - "A Course of Differential Geometry and Topology". It develops everything up from $\mathbb{R}^n$, curves and surfaces to arrive at smooth manifolds and LOTS of examples (Lie groups, classification of surfaces, etc). It is also filled with LOTS of figures and classic drawings of every construction giving a very visual and geometric motivation. It even develops Riemannian geometry, de Rham cohomology and variational calculus on manifolds very easily and their explanations are very down to Earth. If you can get a copy of this title for a cheap price (the link above sends you to Amazon marketplace and there are cheap "like new" copies) I think it is worth it. Nevertheless, since its treatment is a bit dated, the kind of algebraic formulation is not used (forget about pullbacks and functors, like Tu or Lee mention), that is why an old fashion geometrical treatment may be very helpful to complement modern titles. In the end, we must not forget that the old masters were much more visual an intuitive than the modern abstract approaches to geometry.
NEW!: the book by Mishchenko/Fomenko, along with its companion of problems and solutions, has been recently typeset and reprinted by Cambridge Scientific Publishers!
If you are interested in learning Algebraic Geometry I recommend the books of my Amazon list. They are in recommended order to learn from the beginning by yourself. In particular, from that list, a quick path to understand basic Algebraic Geometry would be to read Bertrametti et al. "Lectures on Curves, Surfaces and Projective Varieties", Shafarevich's "Basic Algebraic Geometry" vol. 1, 2 and Perrin's "Algebraic Geometry an Introduction". But then you are entering the world of abstract algebra.
If you are interested in Complex Geometry (Kähler, Hodge...) I recommend Moroianu's "Lectures on Kähler Geometry", Ballmann's "Lectures on Kähler Manifolds" and Huybrechts' "Complex Geometry". To connect this with Analysis of Several Complex Variables I recommend trying Fritzsche/Grauert "From Holomorphic Functions to Complex Manifolds" and also Wells' "Differential Analysis on Complex Manifolds". Afterwards, to connect this with algebraic geometry, try, in this order, Miranda's "Algebraic Curves and Riemann Surfaces", Mumford's "Algebraic Geometry - Complex Projective Varieties", Voisin's "Hodge Theory and Complex Algebraic Geometry" vol. 1 and 2, and Griffiths/Harris "Principles of Algebraic Geometry".
You can see their table of contents at Amazon.
Hope this helps... good luck!
A: I'm doing exactly the same thing as you right now.  I'm self-learning differential topology and differential geometry.  To those ends, I really cannot recommend John Lee's "Introduction to Smooth Manifolds" and "Riemannian Manifolds: An Introduction to Curvature" highly enough.  "Smooth Manifolds" covers Stokes Theorem, the de Rham theorem and more, while "Riemnannian Manifolds" covers connections, metrics, etc.
The attention to detail that Lee writes with is so fantastic.  When reading his texts that you know you're learning things the standard way with no omissions.  And of course, the same goes for his proofs.
Plus, the two books are the second and third in a triology (the first being his "Introduction to Topological Manifolds"), so they were really meant to be read in this order.
Of course, I also agree that Guillemin and Pollack, Hirsch, and Milnor are great supplements, and will probably emphasize some of the topological aspects that Lee doesn't go into.
A: Like the other posters, I think Lee's books are fantastic.  I'd start with his Introduction to Smooth Manifolds.  
For differential geometry, I'd go on to his Riemannian Manifolds and then follow up with do Carmo's Riemannian Geometry.  (That's what I did.)
For differential topology, after Lee's Smooth Manifolds, I'd suggest Differential Forms in Algebraic Topology by Bott, Tu and anything (and everything) by Milnor.
A: Guillemin and Pollack's "Differential Topology" is about the friendliest introduction to the subject you could hope for.  It's an excellent non-course book.  Good supplementary books would be Milnor's "Topology from a differentiable viewpoint" (much more terse), and Hirsch's "Differential Topology" (much more elaborate, focusing on the key analytical theorems). 
For differential geometry it's much more of a mixed bag as it really depends on where you want to go.  I've always viewed Ehresmann connections as the fundamental notion of connection.  But it suits my tastes.   But I don't know much in the way of great self-learning differential geometry texts, they're all rather quirky special-interest textbooks or undergraduate-level grab-bags of light topics.  I haven't spent any serious amount of time with the Spivak books so I don't feel comfortable giving any advice on them. 
A: Here is my list of about 60 textbooks and historical works about differential geometry.
differential geometry textbooks
About 50 of these books are 20th or 21st century books which would be useful as introductions to differential geometry. I give some brief indications of the contents and suitability of most of the books in this list.
A: I had seen a mention of this work on Differential Geometry by Theodore Shifrin at UGA giving it great comments mathoverflow.
It's currently a free and legal download. It's an entry level text and the prior responders have put a lot of effort into giving outstanding suggestions. But I thought it might be of interest.
You can download it with the link on Prof. Shifrin's home page:
https://www.math.uga.edu/directory/people/theodore-shifrin
