Differential geometry study materials

I would recommend Lee's book "Introduction to Smooth Manifolds." It's a long book but is comprehensive, has complete proofs, and has lots of exercises.

M. Spivak, A comprehensive introduction to differential geometry, Publish or Perish, Wilmington, DL, 1979 is a very nice, readable book.

If you prefer something shorter, there are two books of M. Do Carmo, 1. Differential geometry of curves and surfaces, and 2. Riemannian geometry.

I recommend an older book, Notes on Differential Geometry by Noel Hicks. What I like about it is that it starts with manifolds embedded in $R^n$, and shows how all of the concepts of differential geometry naturally arise there.