It really depends on what you are looking for. We stayed in Williamsburg for the duration of our trip (about 5 days) to get a feel for it, as lots of people recommended it. It turns out we didn't really like it (nothing bad per se, just not what we're looking for).
We settled on Lower East Side, as it best represents what we're looking for.
To handle the issue of apartment hunting while not living there, we're going to end up just using airbnb to find a monthly rental, move our stuff up and put it in storage (we're moving from DC), and hopefully will be able to find a place easier by being there.
We're going to live in Brooklyn for sure. My girlfriend likes Brighton Beach, and I'm from Russia, originally, so I wouldn't mind living there. We also heard great things about that hipster part of town whose name I forget.
We should probably combine forces somehow.