Many books on special relativity eventually mention that the geometry of spacetime is special because the metric has a signature $(-,+,+,+)$ which is non-Euclidean. I have encountered many ways this makes it different from normal Euclidean geometry, for example, there is more than one null vector.
I want to study the mathematics of this new geometry in order to develop some intuition for it. I understand that the new geometry is called Hyperbolic geometry. Unfortunately, the information I find about that is all about negatively-curved saddles and Poincare disks, etc, which while interesting, seems quite different!
Can someone point to a good resource for learning just the geometry that underlies SR?