Given the number of things to see and do in NYC, and the number of web sites and publications devoted to listing them, this page contains a few important links to assist you in your sightseeing.
Museums: New York has many important museums, among them The Metropolitan Museum of Art, The Museum of Modern Art, and The American Museum of Natural History. Many more are listed here.
Theater: The Times Square theaters are easy to reach from Columbia via the number 1 subway. One listing of shows is here. Discount tickets can be available through the TKTS Discount Booths.
Nightlife: Many options exist here, with the Lincoln Center complex (Metropolitan Opera House, Avery Fischer Hall) easily accessible from Columbia. New York magazine has one list of events for June 22 to 29.