When I first visited NY in 2010 the New Museum left a permanent impression. I love the architecture and the concept of the piled-white-shoe-box look-a-like building at the edge of the Lower East Side neighborhood. The exhibition program is appealingly different, and the bookstore in the foyer is not to be missed!
Best contemporary art museum in NY, have to take a train that goes along the Hudson river (best views during fall season!) / rent a car. This museum only uses natural light, and has pieces from Richard Serra, Judd, Sol Lewitt, Walter de Maria, Robert Smithson, Flavin, all best of the best!
Manhattan's Chinatown is my second favorite neighborhood next to Fort Greene. It's full of people, odors, bars, clubs, cafes, and restaurants. There's always something going on here.