I think for your price range you can live virtually in any neighborhood in and around Raleigh's downtown that you'd want to.
Oakwood, Mordecai, Oakdale, Pilot Mill, Boylan Heights, Brooklyn, Glenwood South, Cameron Park, Oberlin, Cameron Village, Longview, Hayes Barton, University Park, Roanoke Park, Sunset Hills, Budleigh, Fuller Heights, Lakewood, Pullen....the list goes on.
For the really nice historic ones, I'd say start your search with Oakwood, Boylan and Cameron Park, then on to the others.
As others have said, Public Transport here isn't all that, but it is easy to find a good place in your price range within walking distance to just about everything there is to do here.
For more info about what our downtown is like check out downtownraleigh.org and downtownraleigh.com --you can find listings of most services in the area.