When I was first learning, the best book I read, by far, was Bishop & Goldberg, Tensor Analysis on Manifolds. This is the book I really learned from. Concise and crystal clear.
Another good introductory text is Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry.
Spivak, Calculus on Manifolds is also fine for the very basics.
My favorite book is Jost, Riemannian Geometry and Geometric Analysis. But it's not good as a primer.

Introduction to smooth manifolds by Lee is quite good!

You won't go far wrong if you keep in mind the core idea that differential geometry is about making calculus/curve integrals work on non-flat spaces.

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