While many of the best free daytime activities are outdoors, there are TONS of indoor activities in NYC.

There are scores of museums (many free). For art, check out galleries in Chelsea, or museums like the Met, Guggenheim, Whitney, Frick or Brooklyn Museum. For history, check out the Museum of the City of New York, the Museum of the American Indian, the Museum of Natural History, the Tenament Museum, the Holocaust Museum, the Studio Museum in Harlem, Museo del Barrio, and countless other smaller institutions.

Also pick up a copy of Time Out NY when you arrive (published on Tuesdays with listings for Thursday-Wednesday of the next week). It'll have near-complete listings for the hundreds (if not thousands) of daily music (jazz, rock, hip-hop, electronic, world, classical, opera, etc.), film (Hollywood, independent, foreign, revival, avant-garde), theater (Broadway, Off Broadway, Off off Broadway), and other performances going on around the city. Plus there are lectures at the city's many universities and a variety of other random events and exhibits that are open to the public.

And that doesn't even start to get into the restaurants, bars, nightclubs, and sporting events.