I lived in NYC for a few years, so here's a few suggestions:
Museum of Natural History & the Hayden Planetarium (upper West Side); Metropolitan Museum of Art (5th Avenue by Central Park) - worth it alone for the Egyptian exhibit and the Impressionist collection. If it's not too chilly, Central Park has the Zoo and many nice paths to walk (grab a hot dog or two from one of the street vendors - they're delicious). Fifth Avenue is just fun to walk down - some great store windows to look in and the people watching can be fun. Take the subway to get around (don't be afraid - it's much cleaner and safer than it used to be). Downtown, there's the Empire State Building (34th & Fifth), Macy's department store (same area),
Broadway (West Side from 59th St. - Columbus Circle down to Times Square and the Theatre District at 42nd St.) And much more...depends on how much time you have.