I've been reading about tensors recently and trying to wrap my head around the idea. I think I'm starting to get it, but it would be a big help if someone could tell me if the following ideas are correct and correct them if they are wrong:

The tensor product of two tensors V and U with ranks R and S is of rank r+s and contains every element in V times every element in U and the dimension of that tensor product is the dimensions of V for the first r indexes and the dimensions of U for the last s indexes.

A linear transformation of a tensor in a tensor space of rank r with dimensions D1..Dr to a tensor space of rank s with dimiensions E1..Es is the same rank and dimensions as the tensor product of tensors in those two spaces.

Here's is where I really get stuck:

Given a transformation T from space V to U, and a tensor field R as a function of a tensor X. How would you apply that transformation? Is it just T*R(x) or is it T*R(T*X) or is it R(T*x) or T*R(T^-1 * x) or do I need more information to transform a field than I would need to transform a regular tensor?

Any explanations, formulas, or good links would be appreciated.