Well, I've been through Calculus, Vector Calculus, Ordinary and Partial Differential Equations, and Complex Variables. I guess I just want to learn it because I was planning on going on towards Relativistic Physics and Quantum Mechanics and Field Theory.
I second Bishop and Goldber's Tensor Analysis on Manifolds. I'm reading through this book right now, actually, and it has been quite a pleasurable experience. The notation is a bit awkward (he writes f(x) as fx without parentheses, for example) sometimes, but for the most part this is a thoroughly modern book.
I will say, however, that to get the most out of this book you need some basic background in topology. You could make it through this book without knowing much about topology, but I think you'd miss out on a lot of good material concerning the topological peculiarities of various structures studied in the book. You also need to be familiar with some topics from advanced calculus such as the jacobian, the implicit function theorem, the inverse function theorem, and integration on arbitrary-dimensional Euclidean spaces.
Previously I was grappling with Edwards' Advanced Calculus: A Differential Forms Approach (which isn't really about tensors in general but differential forms specifically). This book took too pragmatic an approach for my taste. Maybe I'm insane, but I actually find the modern, abstract definitions easier to understand and use than the old, often physics-based explanations. Eventually I got tired of trying to translate the practical explanations into the abstract currency of modern mathematics, and I got myself a copy Tensor Analysis on Manifolds, which cured all my tensor-analytic ills.