Housing & Neighborhoods
Seattle's central neighborhoods are considered very desirable places to live but tend to be more expensive than more distant locations. The neighborhoods of Capitol Hill, Wallingford, Green Lake, Ravenna, Roosevelt, and Fremont are all great places to live and are a short bus or bike ride from the University, or in the case of Capitol Hill and Roosevelt, 5 minutes on the light rail. Ballard and Queen Anne are a couple of other very nice neighborhoods you might consider (Ballard is an especially popular choice), although they are slightly farther from the university. Lake City and Greenwood have lots of apartments and are slightly less expensive than the other neighborhoods mentioned. Greenwood, in particular, has lots of amenities and is next door to Phinney Ridge, a very nice neighborhood (with very few apartments). They are also a bit farther from UW and, in the case of Lake City, a bit less charming. Northgate is not especially interesting, but it does have a light rail connection to UW that will get you to the U-district in 10 minutes and has lots of apartments. There is plenty of off-campus housing nearby in the University District. It's very convenient, but housing tends to be a bit rundown. There are new on-campus graduate housing options for those who prefer to go that route.
The farther from the city center you go, the lower prices tend to be. Those looking for the least expensive apartments will find them in the north and south end suburbs (not the eastside suburbs - due to Microsoft et al, housing in Bellevue, Kirkland, and Redmond is as expensive or more than central Seattle) but will have a longer commute and less interesting surroundings. If you do opt for less expensive housing in the suburbs, you'll want to choose one that has frequent all-day bus service to the UW. The north King County suburbs of Kenmore and Bothell fit this requirement (all day & evening bus service via #372 metro transit, as well as connections to light rail via the the #522 sound transit bus) and Bothell in particular has lots of amenities. The Snohomish County suburbs of Lynnwood and Mountlake Terrace, while a significant distance from UW (10-12 miles), have good connections to the UW via light rail.
Prices vary, but in Seattle one bedrooms start around $2200 per month and two bedrooms start around $2500...and go much higher. Seattle has become expensive in recent years, but it's still less expensive than the Bay Area, Los Angeles, Boston, and New York. If you're moving here from a midwestern college town, you can anticipate a bit of sticker shock.