Specific to NYC, as I'm pretty much a native, skip the Statue of Liberty and Ellis Island. If a view of Manhattan is what you're after you're going to get that walking to the Exchange Place PATH station in the mornings and back in the evenings, or take the free Staten Island Ferry roundtrip from Battery Park.
Here is a whirlwind suggested day tour route: do a loop of Manhattan island from downtown by the WTC up to Central Park West say to the American Museum of Natural History (between 77th to 81st Streets), then go across Central Park to the Metropolitan Museum of Art on 5th Ave. The Guggenheim is a few blocks up at 88th. If museums aren't your thing, and yeah, you can spend a day in any of these, just using them as a landmarks on the walking tour is enough. The subway is handy to jump from areas to areas, but like many natives from the NYC area, if the weather is cooperative, you get a much better feel for the city by being a pedestrian to the greatest extent possible. Other good areas and things to explore: the Village, the Grand Central Station Planetarium ceiling, Union Square and Washington Square Parks, Chinatown, Little Italy, and the Brooklyn Bridge. Look at a map and see what pops.
Try some local pizza. It will be better than any you've had, as well as a bagel. Another local item not replicated as good anywhere else.