It seems to me that the most important thing to learn when you're a graduate student is how to learn more mathematics. Everything else is detail. So you do what you learned to do as a graduate student (in order of increasing effectiveness, at least for me):
- Read papers and books (I'm actually unable to do this. I fall asleep.)
- Sit in on courses
- Work out problems in books
- Try to work out the details of a paper yourself and refer back to the paper when you get stuck
- Set up working seminars with other people who want to learn the same thing
And I'll repeat what one of the other answers said: ask every dumb question that comes to mind and that you can't figure out the answer to. This can be done in person, by phone, or by email. Or even on MathOverflow.