NY in a day: If you only have a short period of time, I'd suggest trying this tour to get in a general walk through the city, with the help of MTA, the trusty subway system. Take the A or the 1 train to the World Trade Center. After visiting the site, walk down Church Street towards Battery Park and get a great view of the harbor.
Walk over to Clinton Castle. If you'd like to visit the Statue of Liberty, this is where you can buy your tickets. Or...catch the Staten Island Ferry for free instead. You get a great view of the Statue of Liberty and the NY skyline, plus you get to rest your feet for a while!
If you've had time to rest your feet, keep on walking up State street and you'll end up in the Financial District. you can visit Wall Street, touch the bull for good luck, and peek in on the Stock Exchange.
You're probably getting pretty hungry, so you have a few choices: You could head up to Little Italy or over to South Street Seaport.
There are benefits to both. If you go to Little Italy, you can move over towards Chinatown afterwards for some desserts or bubble tea. If you go towards South Street Seaport, you can go from there to a walk over the Brooklyn Bridge. Take the train from either location to Union Square Park. On nice days and weekends, there are often bands who perform for free. Also there's some great shopping in that area. I'd hop on a bus towards Midtown, and see some of the streets from above ground. It helps to orient you in your trek "uptown."
You can always get off at Times Square, but I'd continue up towards Rockefeller Center. 
Check out the scene of the most famous Christmas tree lighting. Across 5th Ave is St. Patrick's Cathedral, reknowned for its beauty. Only a few blocks up and you hit Central Park! This is an attraction in itself, from the quaint bridges to the open Sheep's meadow, there's always something going on. If you're around 67st St by the Great Lawn, there's a rollerskating disco group, who skate like no others. Walk out of the park onto the East Side, and you hit the Museums. Both the Met and the Guggenheim are right here, although you could spend a day in the museums alone. If you are more interested in natural history,walk out on the West Side.
Times Square Area: Take a few pics in Times Square with a variety of backgrounds. Also check out the huge M & M store. Two blocks down, you can visit the Free Public Library, which I remember best from Ghostbusters! Grand Central Station, at 42nd and Park, is another attraction. There are free tours on the hour every day!
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West Side: A few attractions that you could visit: Lincoln Center and Avery Fisher Hall, home of the New York Philharmonic, NY ballet, and City Opera. A few blocks up, on 68th and Columbus, is a second location of the famous bakery, Magnolia.
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