Required Textbook
Carroll, Sean M. Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 2019. ISBN: 9781108488396.
Additional Textbooks
General relativity is a subject that is either blessed or cursed (depending on your point of view) with an abundance of textbooks. Speaking for myself (Hughes), I love none of these textbooks (including the required text), but rather find that all have strong points and weak points. Carroll’s text does a nice job hitting all the main points needed (especially for a 1-semester class), and does so using language and notation that is up-to-date.
There are several other texts that you should be aware of:
Misner, Charles W., Thorne, K. S., & Wheeler, J. A. Gravitation. Princeton University Press, 2017. ISBN: 9780691177793.
Universally known as MTW. A nice reference once you already know GR thoroughly; not so great if you are studying it for the first time. Very good for certain important topics (e.g., spherically symmetric stars, black holes); a few readings will be assigned from this volume. Carrying this textbook around for several weeks is an excellent way to strengthen your lower back.
Schutz, Bernard. A First Course in General Relativity. 2nd Edition. Cambridge University Press, 2009. ISBN: 9780521887052.
Gives very clear and careful introductory discussion of the mathematics that underlies general relativity. Many of the first (foundational) lectures in this class have their roots in Schutz’s discussion. Be sure to use the second edition, which updates and corrects some errors in the first edition. (Schutz tells me that a third edition is in preparation, updating the discussion to take into account the past few years of gravitational-wave discoveries.)
Hartle, James B. Gravity: An Introduction to Einstein's General Relativity. Harlow: Pearson, 2003. ISBN: 9780805386622.
A nice introduction to the subject, with the aim to get to important physical concepts as quickly as possible. As a consequence, Hartle jumps around a bit, deferring the introduction of some important quantities (such as curvature) to rather late in the text. More elementary than appropriate for a graduate course.
Weinberg, Steven. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley India, 2008. ISBN: 9788126517558 .
This textbook was originally published at almost the same time as MTW; as a consequence, several generations of GR students were educated using either MTW or Weinberg. Takes a rather different point of view, trying to avoid becoming enraptured by the notion of geometry. Instead, Weinberg presents GR, as much as possible, as a classical field theory like any other. Time has not been terribly kind to this viewpoint, so this text is now considered somewhat deprecated. Nonetheless, it is beautifully written and very clear. Worth knowing.
Wald, Robert M. General Relativity. University of Chicago Press, 1984. ISBN: 9780226870335.
The GR überbuch; typically the final arbiter of right and wrong in this subject. Quite mathematically sophisticated, and rather terse. A few pedagogical gems are hidden here (e.g., the nicest proof of the Bianchi identity I’ve ever seen).
Poisson, Eric. A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge University Press, 2008. ISBN: 9780521537803.
The focus of this book is the machinery needed for advanced analysis of black holes. I like to use bits and pieces of Eric’s analysis in 8.962.