I learned it quite quickly from Landau&Lifshitz vol. II. It comes to the point without much ado, and it's clearly written including the necessary tensor calculus (Ricci only, no modern Cartan formulation).
Then there's also Misner, Thorne, Wheeler, where you precisely get the more advanced method, and it's written in a way that it's fun. There's just a reprint of the original book, which is even at a fantastic price (in Germany it's just 60EUR, which is really a good price for such a voluminous physics textbook of this outstanding quality of content).
Another good choice, and a good alternative to MTW, because it's emphasizing the physicists' point of view over the geometers' point of view, is Weinberg's Gravitation and Cosmology.
If you are aiming also at cosmology, you should get a very new text since all the books I mentioned above a pretty much outdated given the enormous progress made in the field. Again, there's a book by Weinberg, Cosmology (2008).