To become a expert on general relativity, one really must tackle these classic texts:
Gravitation, C. W. Misner, K. S. Thorne, and J. A. Wheeler (W. H. Freeman, New York, 1973).
General Relativity, R. M. Wald (University of Chicago Press, Chicago, 1984).
Along with these textbooks, you'll want to do lots of problems! This book is a useful supplement:
Problem Book in Relativity and Gravitation, A. Lightman and R. H. Price (Princeton University Press, Princeton, 1975).