There are a number of ski resorts withing a few hours drive of Seattle. For web pages of the resorts and current conditions, go to
http://www.skiwashington.com/conditions.php
Snoqualmie Summit: About a one hour drive from Seattle on I-90, rather basic runs. Can be very crowded on weekends due to ski schools.
Alpental: Also about a one hour drive from Seattle on I-90, mostly intermediate and advanced runs.
Stevens Pass: A large resort with a variety of terrain, about 2.25 hours from Seattle on US 2, but serious traffic jams can occur on this highway. Best route: cross Lake Washington on 520 E, take 405N and then 522 to intersect US 2 in Monroe. 1000' higher than Snoqualmie. No local lodging, except for members of the Mountaineers Club.
Crystal Mountain: The largest and most varied resort nearby, with all levels of terrain, spectacular views of Rainier when clear. Local lodging exists. About 2.25 hours drive. Best route from the UW: 520 E, 405 S, 169 S to Enumclaw, 410 E to Crystal Mtn Rd.
Mt. Baker: The most reliable for having lots of snow with a variety of terrain. About a 3 hour drive, no local lodging exists except for members of the Mountaineers Club. Interesting terrain, great views of Mt. Baker. Take I-5 N past Bellingham (a delightful town), turn E on the Mt Baker highway.
Mission Ridge: A smaller but quite fun ski resort in Eastern Washington. Short lift lines, mostly intermediate and easy advanced runs. Can be very cold and windy. A 3 hour drive, best route is I-90E to 970/97 N to US2 E. Drive through Wenatchee (where one can lodge) and follow signs to Mission Ridge.
Nordic skiing
There are several cross country skiing centers with groomed tracks in the area. The closest is at Snoqualmie Summit East which has a fun 14 mile loop (the Mt. Catherine loop) which lends itself to excellent (and aerobic) skating.
Another area well known for its cross country skiing is the Methow Valley, on the eastern side of the North Cascades. This area was recently featured in a New York Times article.
Where one really starts having fun is with the back country skiing. There is a system of "Sno-Parks", where one can park the car in plowed areas throughout the mountains, provided you have a Sno-Park permit. A convenient place in Seattle to purchase them is REI. There exist a number of books describing back country ski tours in the area, such as:
If you want a beautiful backcountry day trip on a clear day, drive to Paradise Lodge on the south side of Mount Rainier (bring chains!), ski the Stevens Canyon Road to the end of Mazama Ridge, and ski up the ridge.