Marie desJardins has a nice article on Surviving Graduate School that is definitely worth reading.
The top two pieces of advice I would give are:
The most important thing when choosing an advisor is to find someone who will go out of his or her way to help you succeed, not someone who is famous, and not even someone whose research is in the right area.
You need to make the transition from being a mathematics student to being a mathematician. That means thinking of mathematics as an arena where you seek out unsolved problems and obsess over them until you solve them, not as a vast sea of material to be learned. Don't get sidetracked trying to learn everything; that's impossible. Focus on finding an open problem you can solve, and solve it.