I am a math major and I am learning GR by myself right now. For GR, you need tensors (among many other things), and as a result I have gone through SEVERAL tutorials, books, videos etc.

There is ONE main thing I find lacking in all of these sources: computational examples/exercises.

My idea of a "gentle introduction to tensors" would be: motivation, definition and then immediately followed by computational problems (lots of them). Only then I would be comfortable with abstract definitions and proofs (which is my goal ultimately).

Edit: https://www.youtube.com/watch?v=5oeWX3NUhMA&list=PLFeEvEPtX_... is the most amazing lecture on Tensors I have watched so far, by far!