The modern introductory GR textbook I currently like the most is:-
General Relativity: An Introduction for Physicists(Cambridge, 2006).
I strongly recommend that you obtain a copy, to give you a different but related perspective to the lecture notes. I will refer to this book fairly often in lecture notes as
Introduction to General Relativitynotes (free online version, 2013);
Spacetime and Geometry: An Introduction to General Relativity(Pearson, 2004);
Gravity: An Introduction To Einstein's General Relativity(Pearson, 2002).
Introducing Einstein's Relativity(Clarendon, 1992);