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.