How much calculus do you know ? If you're new to calculus, I recommend Courant. If you've been exposed to multivariable calculus, I recommend Spivak's Calculus on Manifolds.
Edit: A broad introduction to multivariable calculus can be found in the very readable textbook "Vector Calculus, Linear Algebra, and Differential Forms (A Unified Approach)" by Hubbard and Hubbard. It has a lot of practical problems and applications to avoid the the lull of increasing abstraction with little computation.
Before going to multivariable calculus or anything to do with general tensors, I recommend a solid grounding in linear algebra. Try "Linear Algebra Done Right" by Sheldon Axler.
Tensors are best studied as part of differential geometry. Only after having mastered linear algebra should you proceed to multivariable calculus and differential geometry. A good start for DG is Spivak's Calculus on Manifolds followed by his "Comprehensive Introduction to Differential Geometry". You will want to know ODEs and PDEs (ordinary and partial differential equations) as well. Get some Dover books on those (very inexpensive).
Darling's "Differential Forms and Connections" is a good companion for a course in DG, but should not be used by itself. A short intro to mechanical manipulation of tensors can be had by using Schaum's Outline of tensor calculus.
There are a myriad of topics associated with the above, such as group theory, topology, modern algebra, and such, but you'll probably find those on your own. :)