Background:
Notes on Introductory Point-Set Topology (Notes by Allen Hatcher)
Primary Resources:
Lecture Notes on Differential Geometry (lecture notes by Eckhard Meinrenken)
Calculus on Manifolds (Textbook by Michael Spivak)
Introduction to Smooth Manifolds (Textbook by John Lee)
Secondary Resources:
Differentiable Manifolds (lecture notes by Eckhard Meinrenken)
Riemannian Geometry (Textbook by John Lee)
Interesting links:
Interesting Manifolds:
Clebsch diagonal cubic surface
Sage Math can be used for computations and plotting.
You can run it through a web browser at: Sage Math Cloud
Or you can download it an run it locally.
As an example, here is how to use Sage to plot Steiner’s surface, and find the critical points and regular values:
Or the intersection of a torus with a plane (along with various regular values, but note of warning: Sage misses some of them!):
Torus-Plane-Intersection [ipynb,html]
Here is an example visualizing various 1 and 2 dimensional tangent spaces:
A great link to an Sage example of the real projective plane.
More examples can be found at Sage Manifolds.