what's your idea of fun? Museums, theater, sports, dancing, bars, historical sites, zoos, beach, parks, etc?
Museums: American Museum of Natural History/Hayden Planetarium; Metropolitan Museum of Art; Museum of Modern Art; The Cloisters (medieval art)
Theater: dozens concentrated in Greenwich Village and around Times Square (the latter tend to be expensive)
Sports: Yankees. Mets; Rangers; Nets; etc
more history than can fit here
zoo in each borough, each with a different specialty