I'm completely biased as I live in Brooklyn (Park Slope), so I think you should stay here the whole time. Truly. It depends on your style, but if you want tons of shopping and nightlife in your neighborhood, then do Williamsburg. If you want a charming, pretty neighborhood, then do Park Slope, Prospect Heights, Carroll Gardens, Fort Greene or Boerum Hill. Those neighborhoods have shopping and nightlife too, but of more of a "let me grab a drink in my Chucks with some friends" vibe whereas Williamsburg is a scene. Either way, more bang for your buck, cooler people, more dynamic and diverse culture and blue sky and trees.
Pro Tip: Transportation is EVERYTHING. Wherever you look at, double check Google maps to make sure it's within a couple blocks of a subway. Even better if it's an express subway. For example, I live by the express Q and B (and the 2/3) on the north end in Park Slope. In Soho in 15 mins. On the south end they have the local F train....whoa. Half hour to the same neighborhood.
Depends upon your age. Williamsburg and Bushwick and Red Hook are a younger scene- perhaps 20-30 years or so. Park Slope, Carroll Gardens, Prospect Heights are a bit older, more sedate but still great with plenty of good restaurants. Any restaurants on 5th ave in the Slope or Smith Street in Carroll Gardens would be a great bet.