This is only my second question on MM. I first learned about MM during a random conversation with strangers at a public park. It's a fascinating concept, just trying to explore by asking this question.
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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.
DG isn’t my forte so hopefully there will be a few other folks to chime in, but I watched this discrete differential geometry lecture series a while back and enjoyed the way he introduced concepts https://youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS&si=jDUAAyHyzhrklXzL