While not a skier school in the tradition of some other places (MSU Bozeman would be an example), UPS and PLU (my alma mater) offer plenty of options for outdoors minded folk.
As Mike mentioned, Snoqualmie's your best bet for skiing from Tacoma. Cheap pass, night skiing, Joe Pope crushing it in the Central park and seriously epic backcountry at Alpental. It's only an hour and change away. Some folks opt for Crystal, which certainly has the terrain covered, but it's more expensive and there's still a large degree of assholery. If you really like time in the car (two hours each way), Stevens Pass offers the best all around skiing in WA--decent elevation to dodge more of the rain, fast lifts, gnarly terrain, decent jibs, the best jumps in the state, and oodles of epic backcountry.
Tacoma itself has a lot going on. If you like to climb, there's Edgeworks (by SR16) and Vertical World (downtown, bouldering mainly) which have student rates. All sorts of tasty food: Frisko Freeze for burgers and grease, Gateway to India for delish indian food, Vuelve a la vida to eat with all the local hispanics. While only worth one visit, the Glass Museum is neat. The Grand Cinema does all sorts of cool stuff and puts on great film regularly. Plus, Seattle is only 45 minutes out or a cheap bus ride.
Point Defiance is a gorgeous, close option for walks or just hanging out on the beach. Mt. Rainier is about two hours to Paradise; tons of hiking and year round skiing up there. Olympic National Park for when you want to get extra wet in the rainforest or on the coast.
One thing to know as well: Bolt Bus just started super cheap rides ($13!!!!) from Seattle to Portland, so you can explore further south pretty cheaply.
Message me if you've got more questions about Tacoma.