Tensors are a critical part of General Relativity and since I couldn't get three pages into a General Relativity book without encountering tensors, unless it was written by Brian Greene and didn't have any equations, Jason and I figured this was the place to start.
Currently we have started with "Tensors, Differential Forms and Variational Principles" by David Lovelock an Hanno Rund, which came highly recommended by my good friend Mark Friesel. The link below will go over the problems from each chapter. This particular page will discuss the relevance of tensors to General Relativity.
I don't have the book yet, so I'll defer this to someone else. I know that we'll want to talk about vectors, covectors, covariant tensors, contravariant tensors, metrics, rotation, subscripts and superscpripts, conversion between covariant and contravariant tensors, einstein notation, commmon tensors such as the stress/strain tensor. There are other things such as parallel transport and such that I'm not sure when/where we'll talk about that. - Jhouse