David, I am grateful for this topic. I have tried to understand tensors for several months now and I still don't understand the vector space version. I realized, though, that I must understand them well if I want to understand general relativity as well as Lie groups and algebras. I am still not able to make simple calculations. So I appreciate your teaching me what you know about the vector space version.
However, I feel that I have gained a bit of intuition. I think the most important thing is that a tensor is characterized by its type (p, q) where the total dimension n = p+q is to be thought of as broken down into a "bottom-up" component of p-dimensions building up space (adding planes) and a "top-down" component of q-dimensions which is dismantling space (removing hyperplanes). I find this picture helpful: https://en.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors#/media/File:Vector_1-form.svg because it makes clear that there are two different ways of looking at space involved. The vector is made from basis elements that are building up the space but those basis elements don't have to be orthogonal. They are contravariant, which means they are defined opposite to the vector to compensate. However, in the picture there are also covectors, which are defined by the normals to those bases, and in that sense they are orthogonal, I suppose. The covectors are covariant. The key point that characterizes a tensor is how many dimensions of each kind it is using. But the "bottom-up" and "top-down" views are dual ways of looking at the whole space.
The covectors are defined as linear functionals which means they map into a field, say, the real numbers R. They are duals to the vectors. But the duals of the covectors would be linear functionals that, when working in finite dimensions, would match the vectors. In the infinite dimensional case it doesn't necessarily work out. But this all suggests to me that for the sake of elegance it is the linear functionals which are actually more natural and should be more fundamental. So the whole view of "vectors" is, I think, perhaps very unhelpful. Also, the vectors and the covectors are distinguished by whether we are "eating" vectors or "spitting them out", and whether we write them as column vectors or as row vectors.
Another bit of intuition I have is that tensors are what is "maximally trivial". That is, mathematicians like to say certain things are "trivial" and so they don't have to explain further. Well, if we think about that concept, then there is a sense that what is "trivial" is linearity. Linearity is the plain vanilla of math. And then linearity gets pushed to different directions such as taking derivatives. Anyways, I think that tensors are the most robust form of "triviality", that is, of linearity. And they are the default way of making sense of a multi-dimensional space.
I think I know just enough to realize that most people don't really understand what are tensors. I know that I don't understand. So I appreciate your question, your reference and your knowledge. Thank you.