I'd like to ask if people can point me towards good books or notes to learn some basic differential geometry. I work in representation theory mostly and have found that sometimes my background is insufficient.
To Kevin's excellent list I would add Guillemin and Pollack's very readable, very friendly introduction that still gets to the essential matters. Read "Malcolm's" review of it in Amazon, I agree with it completely.
Milnor's "Topology from the Differentiable Viewpoint" takes off in a slightly different direction BUT it's short, it's fantastic and it's Milnor (it was also the first book I ever purchased on Amazon!)
I am teaching a graduate differential geometry course focusing on Riemannian geometry and have been looking more carefully at several textbooks, including those by Lee, Tu, Petersen, Gallot et al, Cheeger-Ebin. I also wanted to focus on differential geometry and not differential topology. In particular, I wanted to do global Riemannian geometric theorems, up to at least the Cheeger-Gromoll splitting theorem. So far, I like Petersen's book best.
Also, as it happens, Cheeger is teaching a topics course on Ricci curvature. He is relying on notes he has written, which I can recommend, at least for a nice overview of the subject. They lay the groundwork for his recent work on Ricci curvature. One of them, Degeneration of Riemannian metrics under Ricci curvature bounds, is available on Amazon.
First, follow the advice that a former Harvard math professor used to give his students. He would point to a book or paper and say, "You should know everything in here but don't read it!". My interpretation of this is to look first at only the statements of the definitions and theorems and try to work out the proofs yourself. Peek at the book only as needed.
Second, follow the advice of another former Harvard professor and develop your own notation. Why? Because it appears that each differential geometer and therefore each differential geometry book uses its own notation different from everybody else's. So you'll go nuts, unless you have your own notation and you translate whatever you're reading into your own notation. Of course, this is a natural thing to do, while you're trying to work out your own proof anyway.
Spivack is for me way too verbose and makes easy things look too complicated and difficult.
I love Guillemin and Pollack, but it is just a rewrite for undergraduates of Milnor's "Topology from a Differentiable Viewpoint". And it's really about differential topology (that is the title after all) and not differential geometry.
For a really fast exposition of Riemannian geometry, there's a chapter in Milnor's "Morse Theory" that is a classic. The rest of the book is great, of course.
Another classic that ties in well with Lie groups is Cheeger and Ebin's "Comparison Theorems in Riemannian Geometry".
I'm recommending only older books, because I haven't kept up with all the newer books out there. One that I also really like is "Riemannian Geometry" by Gallot, Hulin, Lafontaine.
And, back in the day, many of us also learned a lot by reading Thurston's notes on 3-manifolds.
For a more analysis-oriented book, check out Aubin's "Some Nonlinear Problems in Riemannian Geometry". He has a book on Riemannian geometry, but I don't know it very well.
One piece of advice: Avoid using local co-ordinates and especially those damn Christoffel symbols. They have no geometric meaning and just get in the way. It is possible to do almost everything without them. The books I've recommended, except possibly Aubin, aim for this.
I'd start with Lee's Introduction to Smooth Manifolds. It covers the basics in a modern, clear and rigorous manner. Topics covered include the basics of smooth manifolds, smooth vector bundles, submersions, immersions, embeddings, Whitney's embedding theorem, differential forms, de Rham cohomology, Lie derivatives, integration on manifolds, Lie groups, and Lie algebras.
After finishing with Lee, I'd move on to Hirsch's Differential Topology. This is more advanced then Lee and leans more towards topology. Also, the proofs are much more brief then those of Lee and Hirsch contains many more typos than Lee. The topics covered include the basics of smooth manifolds, function spaces (odd but welcome for books of this class), transversality, vector bundles, tubular neighborhoods, collars, map degree, intersection numbers, Morse theory, cobordisms, isotopies, and classification of two dimensional surfaces.
These two should get you through the basics. However, if that is not enough, I'd move on to Kosinski's Differential Manifolds which covers the basics of smooth manifolds, submersions, immersions, embeddings, normal bundles, tubular neighborhoods, transversality, foliations, handle presentation theorem, h-cobordism theorem, framed manifolds, and surgery on manifolds.
Once you have seen the basics, Bott and Tu's "Differential Forms in Algebraic Topology", which is one of the great textbooks, might be a nice choice. The best way to solidify your knowledge of differential geometry (or anything!) is to use it, and this book uses differential forms in a very hands-on way to give a clear account of classical algebraic topology. It wouldn't be a good first book in differential geometry, though.
How basic is "basic"?
For the "basic" material I like the book "Introduction to Smooth Manifolds" by John Lee very much. It's very friendly and very accessible and nicely explains the ideas. Spivak's "Comprehensive Introduction to Differential Geometry" is also very nice, especially the newer version with non-ugly typesetting. Warner's book "Foundations of Differentiable Manifolds and Lie Groups" is a bit more advanced and is quite dense compared to Lee and Spivak, but it is also worth looking at, after you become more comfortable with the basic material.
For a basic undergraduate introduction to differential geometry, I'd highly recommend Manfredo Do Carmo's Differential Geometry of Curves and Surfaces. Nicely done and very approachable, and you'd be well prepared to tackle Spivak's books next.
I feel like this needs to be asked: Is there really such a subject as "basic differential geometry?" I don't believe there is.
Consider the following list of standard topics in "differential geometry" that are, depending on the prof's research interests, either absolutely essential or not covered at all in an intro grad course: partitions of unity, differential forms and Stokes Theorem, Frobenius Theorem, basic Riemannian geometry, fiber bundles, index of a vector field, Lie groups, etc. To wit, if you look at many of the texts listed by other posters, many of them have essentially NO overlap. Just about the only thing that is indisputably a necessary topic in a differential geometry course is the definition of a smooth manifold (and maybe the inverse/implicit function theorem, but probably not). Ironically, even the definition of a smooth manifold is not even a point of intersection, since one may choose to work purely with submanifolds of R^n.
In short, if someone wants to learn some differential geometry, one first has to decide "what kind" or for "what purpose."
I would recommend Barrett O'Neill's Semi-Riemannian Geometry. It's very good. The latter chapters concern general relativity, but the earlier chapters are purely mathematical and contain lots of nice differential geometry.
Here is my list of about 60 books and historical works about differential geometry.
About 50 of these books are 20th or 21st century books which would be useful as introductions to differential geometry at this point in time. In this list, I give some brief indications of the contents and suitability of most of these books.
For a very detailed presentation of some topics from the basic theory in the spirit of algebraic geometry, see Brian Conrad's handouts.
Noel J.Hicks lectures in differential geometry http://www.maths.ed.ac.uk/~aar/papers/hicks.pdf
I would strongly recommend 'An introduction to manifolds' and 'Differential geometry' by Loring Tu. Compared to most other books mentioned, these are recently published. Having gone through both of them, I can vouch for the clarity of presentation and readability. Required prerequisites are minimal, and the proofs are well spelt out making these suitable for self study. The exercises are nice too.
I would like to suggest the three volume set by Dubrovin, Fomenko, Novikov (Modern Geometry--Methods and Applications) as a supplementary reference. They have a somewhat unique style and approach to the subject. The first volume begins with surfaces, the second volume goes on to manifolds. They give examples from physics along the way, which some may find interesting/useful.
I also like the chatty, informal style of M. Berger. He doesn't shy away from giving informal descriptions of ideas and motivations behind definitions. Perhaps most books try to do this, but Berger is particularly generous with it, and good at it, in my opinion. I have his book A Panoramic View of Riemannian Geometry in mind--this may not be the best place to learn about differential geometry for the first time, but I think some of his insights/comments would be useful even for beginners (see, e.g., pp 143-151). I think he has other, more elementary books on geometry but I don't have the references right now.
"Lectures on Differential Geometry" by Chern, Chen, and Lam is an excellent book, and one which truly addresses differential geometry rather than differential topology alone.
Another book I find pretty readable is Modern Differential Geometry for Physicists by Chris Isham. Don't worry about the "physicists" bit in the title, the proofs are not missing there :)
S. Sternberg's recent Curvature in Mathematics and Physics is at about the same level as some of the other suggestions, but includes some extra material hard to find in textbooks at this level. It's also available from Dover, so quite inexpensive.
I am currently using Elementary Differential Geometry by Barrett O'Neill. So far it has been a good reference so long as you have a good grasp what was taught in most linear algebra courses. I will have to update this as the course continues an compare it to the other texts mentioned above.
I would like to add the following notes by Nigel Hitchin: https://people.maths.ox.ac.uk/hitchin/files/LectureNotes/Differentiable_manifolds/manifolds2014.pdf