Tensor Calculus
Solution 1:
DROP EVERYTHING AND GO STUDY LINEAR ALGEBRA
If you haven't taken an advanced linear algebra class, dealing not just with matrices and row reduction, but with vectors, bases, and linear maps, do that. The key to understanding tensor calculus at a deep level begins with understanding linear and multilinear functions between vector spaces. Once linear maps, multilinear maps, tensor products of spaces, etc., are clear to you, come back to this answer.
(Also, as a bonus, deeply understanding linear algebra will also make you understand calculus much better as well.)
Have you studied linear algebra now? Good. The intuition behind tensor calculus is that we can construct tensor fields smoothly varying from point to point. At every point of a manifold (or Euclidean space, if you prefer) we can conceptualize the vector space of velocities through that point. Once we have a vector space, we have its dual, and from the space and its dual, we construct all sorts of tensor spaces. A tensor field is just one such tensor at every point that varies in a differentiable fashion across the manifold.
Let's make a concrete example. You're an EE student, hopefully you'll forgive me if I use a concept from mechanical engineering. Consider a voluminous body with internal stresses. Fix a point. Given a vector $v$ at that point, the stress tensor $\sigma$ produces the stress vector acting on the plane perpendicular to $v$ through that point. It is a tensor because it does so in a linear fashion, at each point mapping a vector to another vector.
If you're interested in general relativity and differential geometry, consider also picking up some differential geometry textbooks. I recommend Semi-Riemannian Geometry, with Applications to Relativity by Barrett O'Neill. (As a plus, if by then your linear algebra is rusty, the first chapter is devoted to the basics of multilinear algebra and tensor mechanics.) You might start by working through his undergraduate curves & surfaces book, Elementary Differential Geometry.