What subject (suggest book titles, etc.) should I study to get a clear grasping of hypersurfaces, 2-surfaces, and integration on them, mostly in special relativity (I'm not messing with general relativity yet).
As others mentioned, special relativity (by definition really) doesn't have anything to do with curved surfaces! Special relativity has a particular metric (minkowski metric) which has no curvature. If your interested in manifolds (particularly integration on them, since integration in minkowski space is pretty trivial) and things like that, you really getting into general relativity.
Although I am sure many people will have pedagogical arguments against the following approach, I will say it anyway... You don't really need to do an in depth study of mathematical SR to do GR. Really once you understand the basic properties of minkowski space, know what a lorenz transformation is, understand things like length contraction, etc, you really should just jump right into GR. One sorta friendly way to go about it is apply SR to E&M (covariant formalization) as a way to play around with tensors a little.
As far as recommendations of what books to get, take a look at Sean Carroll's notes (http://preposterousuniverse.com/grnotes/) and if you like them get his book too! Perhaps also looking at the first two sections will give you a better idea of what you are looking for.
EDIT: After commenting, I thought I would also just post a link to O'Neills Amazon Page. His books are probably more mathematical than what you are looking for, but they could be very useful to someone else.