Printed Matter—organizer of the NY and LA Art Book Fairs—is a non-profit supporting the work of artists and independent publishers through the circulation of their books. At their store in Chelsea, you can shop an insanely diverse selection of artists’ books, zines, posters, and more. They’re also always hosting awesome in-store exhibitions and events, from book launches and talks to lectures and performances.
You can't say you've seen NYC until you've seen it from a boat. Hop on the ferry from anywhere to anywhere and you'll see what I mean. Not only will you see excellent old buildings and factories--the inner workings of the old industrial city--but you'll head out under working bridges and see the skyscrapers in all their towering glory. And once you've done it, do it again, at night.
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.
Luhring Augustine did a good job of converting an old warehouse/storage facility into a blue chip commercial art gallery that stages four shows of contemporary art per year. Which means you can catch museum quality shows in an unlikely spot, away from all the Chelsea pomp, and it actually has friendly staff, instead of a cold gallerina sat behind a desk pretending to be on the phone.