Five years ago Kip Thorne recommended Hartle's book "Gravity: An Introduction to Einstein's General Relativity", and it is a wonderful introduction to General Relativity. (I took calculus, linear algebra, advanced linear algebra, differential equations, real analysis, complex analysis, differential geometry, and Fourier analysis before taking the graduate course on General Relativity at UC Davis)
We used Carroll's book on General Relativity, which is more or less his lecture notes which are freely available online at http://preposterousuniverse.com/grnotes/
As far as understanding Einstein's field equations, a wonderful reference by John C. Baez and Emory F. Bunn: The Meaning of Einstein's Equation on Baez's webpage.
Dr Carlip recommended R. A. D'Inverno's Introducing Einstein's Relativity (Oxford University Press, Oxford, 1992) to the graduate physics class.
Dr. Joseph Biello recommended B. F. Schutz's A First Course in General Relativity (Cambridge University Press, Cambridge, 1985) as an introductory text, then follow it up with Robert Wald's General Relativity (University Of Chicago Press, First Edition edition, 1984). I don't know about this, because Wald himself admitted that he didn't give justice to the Lagrangian or Hamiltonian formalism for General Relativity.
When thinking about black holes, or any singularity, the standard reference on this subject is Hawking and Ellis' The Large Scale Structure of Space-Time (Cambridge University Press, 1975).
Everyone says "Misner, Thorne, and Wheeler" and I am inclined to agree. If you are serious about studying General Relativity, you have to read through this at least once. It is the Bible of General Relativity, although there are two opinions people have about it: they love it, or hate it. Those that hate it would object to my opinions...
When you want to start getting into quantum gravity or numerical relativity (i.e., using the computer to solve Einstein's field equations), you need to learn the ADM formalism. There are a few books on this.
Peter Peldan wrote a review article, "Actions for Gravity, with Generalizations" arXiv:gr-qc/9305011v1 which is a great review of all the many different types of variational ways to obtain General Relativity.
Bojowald just wrote a great book, Canonical Gravity and Applications, which is a fabulous introduction.
Poisson wrote a book on a lot of folklore topics in general relativity, which is based on his lecture notes (freely available online at http://www.physics.uoguelph.ca/poisson/research/agr.pdf). In fact, the only difference I can find is that the pdf doesn't have the index or table of contents, but everything else seems identical.
For numerical relativity, Baumgarte and Shapiro's Numerical Relativity: Solving Einstein Equations on the Computer is another fantastic reference. The only disadvantage is their index system is confusing.
As far as the mathematics, there are many books out there on differential geometry. The graduate course sequence on differential geometry at UC Davis used Manfredo P. do Carmo's Riemannian Geometry
However, many of them kind of ignore the intricacies of Lorentzian manifolds. Arthur L Besse's Einstein Manifolds (Springer, 2007) is a wonderful reference for this subject. (Arthur L Besse is a pseudonym for a group of mathematicians inspired by the Bourbaki group, and works primarily on differential geometry.)