To get a good grounding in tensors, I'm reading the book *The Absolute Differential Calculus (Calculus of Tensors) (Dover Books on Mathematics) Paperback by Tullio Levi-Civita. I'll then move on to a modern book and compare the approaches.
So far, I've been pleasantly surprised by how clear this book is, even if it's hardly a page turner. But I would still like to know in advance how the definition of a tensor in this book differs, if at all, from its modern definition.