General Relativity

Getting started:

There are various textbooks / course notes for general relativity at various levels. Pick the one that's right for you!

Note that people in your department may already have these books, so feel free to ask around!

Black Holes and Time Warps -- this is a non-technical introduction to the beauty of general relativity and black holes. If you are just getting started and want a non-mathematical but conceptually deep introduction, read this book :)
Hartle -- this is perhaps the "friendliest" introduction, and is great for an undergrad-level course
Carroll -- this is a great geometry-focused introduction to general relativity, for a beginning grad course. It's very nicely organized, and relatively short compared to other books.
Tong -- this is a set of lecture notes (if you don't want to buy a book), that's very close to Carroll in scope / flavor.
Misner, Thorne, and Wheeler (MTW) -- this is somewhere between Hartle and Carroll, but is very long. It's useful in the sense that you can use it as a resource for so many things.
Wald -- this takes a much more mathematical look at general relativity. It's very rigorous, but doesn't have as many examples from astrophysics.

Poisson and Will, Gravity: Pedagogical introduction to post-Newtonian theory
Penrose and Rindler, Spinors and Space-time (2 volumes): Thorough text on the spinor approach, necessary for understanding Newman-Penrose, GHP, and asymptotia
Stewart, Advanced General Relativity: More concise than Penrose+Rindler, with a somewhat similar scope
Stephani et al., Exact solutions of Einstein's field equations: Covering topics not seen in other texts such as algebraic classifications (e.g. Petrov and Segre) and symmetries; and with a thorough classification of exact solutions known to the authors at time of publication
Griffiths and Podolsky, Exact Space-times in Einstein's General Relativity: Fewer exact solutions than Stephani+, but each one is covered more pedagogically and in greater depth