It seems as though rents have gone up over the last 1-2 years. Check Craigslist for neighborhood-specific info, but typically in Capitol Hill, Wallingford, U-District, Ravenna (neighborhoods bordering UW), studios are $600-800, 1 bedrooms almost always $750-1100. If you need an especially cheap place, those do pop up, especially in the U-District. And of course house sharing is always cheaper. I've found all my places off of Craigslist and don't know of better ways, other than driving around and looking for signs. If you don't have a car, I'd recommend neighborhoods that border the University District. As for the U-District itself, some people like living there and there are certainly better/worse parts to it. I wouldn't recommend living there if you have a car (break ins). Also, the area just north of campus is Greek Row. Noisy parties, etc. Avoid if this kind of thing bothers you.
Capitol Hill is very conveinent- #43 and 49 buses go from there straight to the school. It's a big neighborhood. Parts closer to Broadway and I-5 are louder/limited parking/more nightlife/pros and cons to this. Further east is (12th-23rd) is generally quieter. Wallingford, to the west of UW, is also quiet and safe (#44 goes from there to UW). Neighborhoods to the North, like Northgate, tend to be cheaper. Further west, but still within 1 bus ride of campus, are Ballard, Fremont, and Greenlake. These are all nice neighborhoods, with plenty of nice restaurants, grocery stores, etc. As with any city, there is crime here so take caution when walking around at night. Certain areas of the Central District (south of Capitol Hill) are dicey and the same goes for other further south neighborhoods (Beacon Hill, etc).
Seattle is great in terms of outdoor activities- there are some beautiful places within a 1-2 hour drive of here (Deception Pass, Olympic Nat'l Forest, North Bend, to name just a few). Great area for kayaking, snowshoeing, hiking, etc. And many gorgeous parks close to UW. Yes, it does rain a lot but I think what gets to people more than the rain is just the general lack of sun. Do buy a good rain coat and keep in mind that a few days out of the year it may snow and be quite cold. And even a couple inches of snow is enough to shut down transit and make commutes horrible!
I'm not a big nightlife person, but there are a ton of clubs and shows in Capitol Hill, Belltown, Pioneer Square (check thestranger.com for listings). There are fantastic restaurants (see http://www.urbanspoon.com/c/1/Seattle-restaurants.html for details) and a surprisingly active literary scene. (Elliott Bay, in Pioneer Square, is a fantastic bookstore with many readings/book groups.) Phenomenal Ethiopean and Vietnamese food.