This won't necessarily tell you how to solve Einstein's equations, but it will tell you what they are.
Solving the equations is much harder, especially in their full non-linear form. Wald has a chapter on "Methods of solving Einstein's equation" in his advanced topics section, for instance. This isn't the sort of thing you are going to pick up by reading a few papers, IMO. Reading the chapter in Wald, for instance, assuming you have or can acquire the math to read it, is more likely to give you an idea of how difficult the problem is rather than to allow you to come up with solutions.
(If you look at some of the book recommendation threads, you'll see that a simpler introduction to GR such as Schutz, DInverno, etc. is recommended before tackling Wald)
Wald considers the case of axis-symmetric time invariant solutions, for instance, simplifies the problem enormously with many dense pages of tensor manipulations, only to mention that most solutions in this category have not come from a direct attack on these simplified equations!
As far as finding tutorial or pedagogical papers about GR on xxx.lanl.gov goes, you can try the results of the following search:
You'll have to be a bit careful about what papers you trust and how much though - lanl.gov now has an endorsement system, but the website is geared for the expert who can evaluate papers himself and wants rapid distribution, and this endorsement system was not always in place but arose when the problem of "crank" submissions became acute.
Thus papers that appear on this website are not necessarily peer-reviewed, though many of the papers have been published and thus have been through the peer review process.
Being published is no guarantee of corectness or widespread acceptance either, but it helps. (A brief glimpse through the list shows a paper by Kaluber on the rotating disk which was published but which I would not recommend, for instance, the paper by Tartaglia http://arxiv.org/abs/gr-qc/9805089 is in my opinion much better. )