This really should be in regionals buuuuuuuuuuut I feel obligated to respond.
UW is so much fun! This is my third year here, and there are a lot of people that are passionate about the outdoors and skiing, not to mention its an awesome school. Definitely expensive if you're out of state, but the resources available to students here are nuts! So many opportunities. Not to mention the campus is absolutely beautiful.
I would say most of the student population skis at either Stevens or Snoqualmie/Alpental. If you won't have a car, I'd plan on skiing at one of those resorts because it'll be easier to get rides.
There is a lot of accessible road biking throughout the city, and really rad mountain biking not too far outside of the city.
Climbing scene is pretty hip - be prepared to get a stone gardens membership and grow a beard. Really big, challenging climbing gyms here though, stone gardens especially. I have heard good things about seattle bouldering project too. There is a lot of great outdoor sport climbing on the east side of the mountains, but definitely some opportunities to boulder and sport on the west side too.