Kip Thorne and Roger Blandford have an excellent introduction to tensors in their lecture notes on the Applications of Classical Mechanics. See sections 1.3 and 1.5:
They emphasize the geometrical viewpoint in which tensors are just linear functions which map vectors onto vectors via the dot product. (Or more generally a rank-n tensor is a linear function which maps a vector onto a rank n-1 tensor via the dot product.) Throughout the entire course they eschew coordinate systems as much as possible since that tends to obscure the underlying math and physics.
I found their introduction to tensors to be the most helpful I had ever read. Realizing that tensors are (or can be considered) functions that map vectors onto vectors was a big conceptual shift for me.
Yes this paper is excellent. Far more comprehensible (to me) than the link at the top of this discussion. I guess it all depends on what you are already familiar with. I was familiar with vectors and functions of vectors, and the concept of a linear function. Therefore, "A rank-n tensor T is, by definition, a real-valued, linear function of n vectors." is a much better starting point for me.
http://www.pma.caltech.edu/Courses/ph136/yr2012/1201.1.K.pdf
They emphasize the geometrical viewpoint in which tensors are just linear functions which map vectors onto vectors via the dot product. (Or more generally a rank-n tensor is a linear function which maps a vector onto a rank n-1 tensor via the dot product.) Throughout the entire course they eschew coordinate systems as much as possible since that tends to obscure the underlying math and physics.