Are you looking for a text on the mathematics or for a text on relativity. Tensors and differential geometry is covered in many basic books (including mine), but they will generally not discuss the specifics of Lorentzian manifolds but instead keep to Riemannian geometry (where the metric is positive definite). The generalisation to pseudo metrics is rather straight forward though and any relativity text should discuss it.
For a GR intro, I really liked Foster and Nightingale's "A Short Course in General Relativity" because it progresses very naturally like this:
1. Curvilinear coordinate systems in flat space (e.g. polar or spherical)
2. Curvilinear coordinate systems for a curved surface in flat space (e.g. a spherical surface)
3. Specialize to remaining in that curved surface, and now you have a curved 2d space. Suddenly we are in GR land.
If you already know what the Jacobian is and how to transform between (x,y) and (r,theta) in flat 2d, then you can quickly get up to speed. If not, he introduces these things, and you'll just have to move a little more slowly.
That depends on how you view it. I will not go into differential forms, but give the old answer. Its as sort of a generalized matrix of numbers that has certain transformation properties. It comes in two general types contravarient and covarient, and even mixed. Like matrices in linear algebra the actual numbers depend on the 'basis' of the vectors it is transforming. But the modern view is like a linear transformation in linear algebra, the actual numbers in a particular basis is not that important - the 'abstract' properties of the transformation is whats is mostly considered.
The only real way to understand them is to go through the literature such as what I posted.