I've lived in Seattle for over six years now and I've tasted all of the neighborhoods. I may be a little on the picky side, but when it comes to convenience, tranquility, aesthetic beauty, and the quality of living here's the "block" that I've come up with: Capitol Hill, North of John St, South of Roanoke, East of Broadway, and West of 23rd Ave E. This area is, in my extremely selective opinion, the most ideal place to live in Seattle; whether you're a professional or a graduate student.
You can live in a calm/peaceful neighborhood, and be just a few blocks away from grocery stores, coffee shops, and a slew of dining options without a bunch of noise. From minimal amenities and sq. footage on up, a one bedroom apartment can range from $800 - $1,800 and a two bedroom is about $1,000 - $2,000. I have found some good deals since I've been here and have lived in some pretty decent places. Currently I live in a one bedroom (within that "block") for $1,375 a month - 800 sq. ft, balcony (which are hard to come by in this area), parking, and it's in good condition.
People around this area are very friendly and open-minded, but it doesn't matter if you're a social butterfly or a sociopath. The overall vibe is live and let live.
Even though most other neighborhoods in Seattle are cheaper to live in they tend be noisy/city-like, except for Queen Anne, but Queen Anne is about a 20-25 minute commute to UW and it's as expensive as Capitol HIll.
I hope this helps a little.
P.S. The Central District sucks.