Reading list for basic differential geometry?

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.