You should have no problem finding a 2 bedroom apartment in Seattle for $1200 or less. Most neighborhoods in Seattle are nice, but where you want to go definitely depends on where you find work, since the traffic may make even a 5 mile drive a pain in the ass.

Microsoft and Nintendo are on the Eastside, across Lake Washington, in the Redmond and Bellevue area. If you get a job there, stay on the Eastside to avoid the worst commute in the Seattle area. Unfortunately, rents are also higher over there.

If you get a tech job in Seattle proper:

Adobe, Big Fish Games, PopCap Games etc., look for housing in Fremont, Queen Anne or Ballard. For Amazon, look for housing on Beacon Hill or Capitol Hill. Capitol Hill is the most densely populated, so rent is a little more expensive and spaces smaller, but the night life is active and is an epicenter for the gay community in Seattle.

Other neighborhoods that are quieter but still awesome include: Wallingford, Greenwood, Wedgewood, Ravenna.

Neighborhoods I don't recommend are Belltown (great bar scene but expensive to live there, and until recently, it used to be low income housing, now yuppies are moving in and the class war is raging). White Center and South Park are getting nicer but still pretty ghetto, with a higher than average crime rate. Cheap housing can be found in the University District (U-District) but it's crap housing for college students and sleep will be hard to come by.