I am reading Sean Carroll's book on GR and have read the first two chapters, which are on manifolds and differential geometry. However, there are only 12 problems for both chapters. In fact, there seem to be few problems for each chapter throughout the textbook. Hence, I wish for a recommendation on a book on general relativity with lots of problems. The book should be mathematically and conceptually advanced, and have plenty of problems (30-50 problems per chapter). The book preferably should either be free or of low cost, because I am self studying general relativity and don't have too many financial recourses. So in brief: what books are there that cover general relativity from the very beginning to cosmology with tons of problems?
The "Problem book in relativity and gravitation" is very good, and written by some well-known relativists. It's got a pretty broad variety of questions, along with solutions. It is a little on the old side, but many of the problems are just as relevant today.
I would get Wald. That's the standard text for the field. It has a small number of problems, but they're very good. I would recommend downloading homework problems from other schools, for example MIT opencourseware. It seems to be the trend that for GR the most popular textbooks are not problem-heavy.
One neat book that is dedicated to problems is the Problem book on relativity and gravitation: http://www.amazon.com/Problem-Book-Relativity-Gravitation-Lightman/dp/069108162X (I'm sure a copy can be found cheaper than this).