On Learning Tensor Calculus I am highly intrigued in knowing what tensors are, but I don't really know where to start with respect to initiative and looking for an appropriate textbook.
I have taken differential equations, multivariable calculus, linear algebra, and plan to take topology next semester. Since Summer break is right at the corner, I was wondering if anyone would tell me if my mathematical background is able to handle tensor calculus, or if I need to know other subjects in order to be ready to learn about tensor. Any and all help would be appreciated.

For a more mathematically flavored introduction, consider studying:


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*Vector Calculus, Linear Algebra, and Differential Forms, by Hubbard and Hubbard

*Semi-Riemannian Geometry with an Introduction to Relativity, by O'Neill

*A good advanced linear algebra book. This should be one that focuses on abstract vector spaces and multilinear algebra, not Gaussian elimination. I hear the one by Roman is good. 
The first one is an introduction to multilinear algebra and its relationship with calculus, pitched at undergraduate students. The second is a book on the geometry required for relativity, but the first two chapters are a relatively rigorous, if perhaps terse, introduction to the linear algebra required to do tensor analysis on a manifold. 
In my own experience, when I was at your stage, I had seen some tensor analysis and had no idea what was going on. It wasn't until I read O'Neill's book pretty carefully that the subject "clicked" for me.
