How to understand the geometry of bilinear forms that are not positive-definite?

Question

I simply cannot find a good resource that explains intuitively how to understand the geometry that is induced on a vector space when the bilinear form is not positive-definite. In the ordinary Euclidean case, we have nice formulas that give us the length of vectors and the angle between them, but apparently these notions break down in the absence of positive-definiteness. How should I try to visualize these strange geometries? What sort of picture should I keep in my mind when I try to imagine, for instance, a symplectic form? Any thorough explanation or reference to an adequate resource would be greatly appreciated.