Mathematica notebook will compute Christoffel symbols,
Riemann, Ricci, and Einstein tensors
for any metric you specify.
Lecture Notes on General Relativity.
The notes do not cover special relativity, and do not get as far as rotating black holes (the Kerr metric).
However, Blau does an incomparable job of exploring
the various subtle features of general relativity
from many complementary points of view.
If you ask yourself, for example,
why does the Riemann tensor have the symmetries it has,
Blau will give you not one but several answers,
among which you will probably find an answer that fits the deep question you had in mind.
Ideal for the graduate student seeking to learn and to understand GR.