A comment to my last post asked for advice for people interested in getting into complexity research. So here it is. Keep in mind that I'm still a grad student -- for advice from more experienced researchers, read Lance's earlier post, and this essay by physicist Steven Weinberg.
I think the key is to start doing creative original research right away. My first year at Berkeley, I took three courses a semester, hoping to prepare by stuffing my brain with knowledge. This was a mistake. Take as few courses as you can get away with, besides directly relevant ones like complexity theory. Learn what you need to know while doing research, not beforehand.
This approach has two advantages. First, you never know what you need to know until you need to know it. Not even Einstein could have predicted as a student that he'd need differential geometry to invent general relativity. And second, you don't really understand anything unless you have a personal stake in it -- meaning that you discovered it, rediscovered it, extended it, applied it, tested it, implemented it, reinterpreted it, explained it to others, etc. This the reason most students forget everything in a course right after the exam. (As Feynman said, "what I cannot create, I do not understand.")
So then, how do you do original research? By throwing your entire life into it. Many researchers play piano, go to clubs, sail, etc., but if they're any good they probably think about research while they're doing these things. I know grad students who never suffer the indignity of working late into the night. They go surfing with friends every weekend and are constantly away on road trips. At this rate, they'll enjoy life more than I will but won't be successful researchers.
I can't offer any advice on research topics, other than to solve the open problems listed in my papers. Blanket advice is difficult because your research ought to be intimately connected to who you are as an individual. Lance suggests leafing through conference proceedings until you find what excites you, while Weinberg suggests getting involved in the "messes" that nobody understands. As for me, I like to start with physical or philosophical questions (can we assign any meaning to "the past" besides memories and records in the present? is there a theory that agrees with quantum mechanics on all experiments to date but that wouldn't allow quantum computation? why should we expect information content to be proportional to area rather than volume?), and then look for related questions that can be addressed using complexity theory. But I don't know if anyone else works that way.