If you're an applied math major, I'd suggest your focus should be on learning about as many markets/instruments as you can... the idea is to build experience, so that *eventually* (a decade from now?) you'll be able to use your analytical background as your edge. I don't think the scalping/daytrading game makes much sense; you wouldn't have an edge over the unemployed high school dropout sitting in mom's basement.
I think UW has a computational finance master's program, don't they? I'd make that your goal.