You could pick a neighborhood and just walk around - you will probably find plenty to shoot, both in Manhattan and Brooklyn (since you'll be staying in these two boroughs). Recent neighborhoods that I've explored that come to mind are Greenwich Village, Chelsea, Tribeca and Coney Island. The first three are in Manhattan, while the last one is in Brooklyn. This is just to give you some ideas. Of course, there's a whole lot more! I find Greenwich Village very bohemian - lots of food places, and live music (the Blue Note is a famous jazz club in this neighborhood). Coney Island has a boardwalk with lots of interesting things to photograph - it is closed for the season now, so only the locals are around.
Good luck, and have fun!