Resources for properly developing a modern understanding tensors I am currently learning about tensors as they come up in the mathematics behind continuum mechanics. 
I was fairly disappointed with my initial foray into tensors, as presented in the book Classical and Computational Solid Mechanics, by Fung and Tong. That experience caused me to write up the following question: I feel that (physics) notation for tensor calculus is awful. Are there any alternative notations worth looking into?
As you can see, I got an interesting answer, recommending that I read through the following paper: Tensor Decompositions and Applications. However, that text is still beyond me at the moment, as it assumes a fair bit of background knowledge on tensors. 
After doing some researching here, I found the following answers:


*

*Suggestion for good notation - abstract index notation

*Harmful heuristics one should avoid - "A tensor is a multidimensional array of numbers that transforms in the following way under a change of coordinates..."
The above heuristic is exactly what is presented in Fung and Tong's book. I would really like to understand tensors properly. I want to avoid harmful heuristics, and unnecessarily tedious notation. I would like to be able to understand for instance, the following sentence (touted as being the "real definition of tensors" according to one of the comments):

Let M be a smooth manifold; let FM be the space of smooth functions on M, let VM be the space of smooth vector fields on M, and let VM be the space of smooth covector fields on M. A (k,l) tensor is a multilinear map from (VM)^k x (VM)^l to FM.

What resources should I pick up in order to begin?
 A: Spivak's book A Comprehensive Introduction to Differential Geometry, Vol. 1, has a great introduction to tensors (see chapters 1 to 4).
A: i'm not familiar wich properties of tensors are essential to physicists, but if you havn't any intuition what tensor products are and you need to overcome obstacles provided by the (unfamiliar) notation: i encourage you to have a look at federico ardila's video lecture on Hopf Algebras. Especially Lecture 4, wich is called "Fun with Tensors". It's a really gentle and example based introduction to the notation and he explains the fundamental concepts of tensor products really well while assuming only a bit of linear algebra. It's not very far though, it might help you get an intuition about how tensors work, and what type of questions you should ask yourself, while working with tensors.
you can find the videos on youtube:
http://www.youtube.com/playlist?list=PL-XzhVrXIVeRLeezwY9h4M68k6yB3yOo-
A: I have not explored this yet in great detail, but perhaps one of the best way to build an understanding of tensors might be found in geometric algebra. See: http://www.mrao.cam.ac.uk/~clifford/publications/ps/chris_thesis.pdf
