I've been reading Fleisch's "A Student's Guide to Vectors and Tensors" as a self-study, and watched this helpful video, also by Fleisch. Suddenly co-vectors and one-forms make more sense than they did when I tried to learn the from Schutz's GR book many years ago.
Especially in the video, Fleisch describes one way of viewing a second-rank tensor; as an object whose individual components relate the input and output components of two sets basis vectors.
In Chapter 4 of his book, Fleisch draws careful attention to two different ways that second rank tensors can be used to relate vectors; they can be used to change a vector, or to change the coordinate representation of a vector.
So, in a classical rigid-body problem, a second-rank tensor is used to create a new vector. The moment-of-inertia tensor operates on an angular velocity vector expressed in terms of a set of basis vectors, and returns an angular momentum vector, which is a different vector that is expressed in terms of the input basis-vectors.
In a change of co-ordinates problem, on the other hand, a second rank tensor operates on a vector expressed in terms of a set of basis vectors, and returns the same vector expressed in terms of a different set of basis vectors.
Even though different things are happening in the two problem types, it is obvious in both cases that the fundamental definition of the problem requires two sets of basis vectors, one used to describe the input and one used to describe the output.
Now, in section 5.5 of his book, Fleisch describes the metric tensor, and describes its function as "allow[ing] you to define fundamental quantities such as lengths and angles in a consistent manner at different locations".
My question is this: Why does this require two sets of basis vectors? If I just want to measure a distance, not change the vector that describes that distance, and not change coordinates, why do I need to involve the second set of basis vectors that a tensor implies? Or, to put the question more concretely, What does the "second set" of basis vectors represent, for example in the Cartesian metric in 3-dimensions?