The Brooklyn Museum is a great place to visit for contemporary and historical exhibitions, showcasing a range of international and local artists as well as its socially engaging and creative program it also hosts a party on the first Friday of every month. Its a great place to see performances (like these two cool cats) and mingle with the locals in the neighbourhood as well as art enthusiasts who come from afar to get down together and party in between the Rodin sculptures.
Apart from Dia Beacon, in Beacon (upstate New York about an hour away from Grand Central Station) and also the Judd Foundation in Soho, this institution is absolutely one of my favourite places to visit and it is located in Long Island City, Queens. It showcases sculpture in the broadest sense and has an incredible building that used to be a workshop that refurbished and fixed subway cars. It has been running since 1928 and when you visit you will definitely see why that is.
The permanent piece called "Meeting" at MoMA PS1 is an etherial experience of light and color and quiet. Sit and look through the cutout in the ceiling and watch the light change both outside and in. You can't tell what you're looking at until maybe a bird flies by.