I wish someone had recommended Paul Renteln's Manifolds, Tensors, and Forms. An Introduction for Mathematicians and Physicists. Unlike many mathematically inclined differential geometry textbooks, it works with an indefinite metric the whole way through.
Chapters one and two aren't very necessary and primarily form a review of linear algebra. It also defines the tensor product. It does contain quite a few gems however. One thing, which isn't quite a gem but in my opinion shows the usefulness of a mathematical point of view: relating raising/lowering the indices to the Riesz representation theorem.
Chapter three is a big chapter, covering the definition of a differentiable manifold, the definition of vectors, the definition of co-vectors, their relation, the definition of differential forms, and lots of applications. I like Renteln's approach because it uses the notion of a differentiable manifold at first, and only brings in geometric manifolds (where a metric is defined and there is a clear map between vectors and covectors) when they are needed. This makes the structure of the theory of differentiable manifolds MUCH clearer. In some physicist definitions of tensors it is never clear what is and what isn't dependent on the metric. It also uses, in my opinion, the best definition of a tensor: A vector is a tangent to a curve. A covector is an element of the dual space of vectors. A tensor is a tensor product of vectors and covectors. None of this "list of quantities that transform as such and such", the transformation laws pop out of Renteln's definitions.
Chapters four and five are on homology and cohomology. I found this stuff really beautiful but skimmed it for expedience - I wanted to get on with it to study general relativity quickly.
Chapter six covers integration on manifolds. It is necessary if you want to study, say, deriving general relativity from the action principles (action is an integral over spacetime), but is mostly optional.
Chapter seven covers vector bundles. This is necessary to understand how tensor fields on differentiable manifolds behave. Again, this section is very general and only brings in the metric when absolutely needed.
Chapter eight is, like chapter three, a very big chapter. It covers all the good stuff - the metric giving rise to the connection/covariant derivative, curvature, torsion, and all of those good covariant derivative identities you need to do any GR! This is the chapter you really need to get started with GR, but it is dependent on chapters three and seven.
Chapter nine is another short one, applying things in previous chapters, but is not so relevant to starting GR.
I spent a whole bunch of time on chapters three, seven, and eight, before jumping into a graduate level general relativity sequence of courses, and found that I was more than well enough prepared.