Only have a day to visit New York City? Here are my suggestions for things to do and see. Most of them are clustered around mid-town Manhattan. And it doesn't have to be expensive. In fact, there's a list of 365 free things to do in NYC. It's the comprehensive directory and includes a handy map.
Begin your visit with taking in the view from the Top of the Rock (50th Street between 5th and 6th Avenues). This observation deck is high atop the famous Rockefeller Center and it provides excellent views of Central Park, the Empire State building, and the rest of Manhattan. Top of the Rock is preferred to the Empire State Building because here you can reserve a time to go up to the top but at the Empire State Building you can't so stand in line longer. While at Rockefeller Center you can see the ice skating rink (in winter) and the beautiful Channel Gardens.
Eight blocks south (a little less than a half-mile) is Bryant Park (on 42nd Street between 5th and 6th Avenues). This relatively small space is possibly the finest urban park in the world. Things to do there include a carousel for the kids, petanque, ping pong, chess, outdoor movies (summer) and ice skating (winter). Eat at the Bryant Park Grill, Cafe or the 'wichcraft stand. There are also free outdoor yoga and tai chi classes. The park is heavily used by New Yorkers at all times. Hang out and enjoy it.
Next to the park is the New York Public Library (main branch, 5th Avenue and 42nd Street). There are three things here you may want to see. The Rose Reading Room on the third floor is one of the most majestic public spaces anywhere. In the children's center is the real original Winnie the Pooh stuffed animal, and his friends. In front of the library are famous statues of two lions. Named Patience and Fortitude by Mayor LaGuardia for the qualities New Yorkers needed to survive the Great Depression, the lions are a great photo opportunity.
Walk two blocks east on 42nd Street and you'll be at Grand Central Terminal (42nd Street and Park Avenue). The main concourse has been used for scenes in many movies and television shows. The architecture is beautiful. When you're outside the Terminal, look up and you'll see the gorgeous art-deco Chrysler Building (it isn't open to tourists). From Grand Central you can take the number 6 subway north to the 59th Street station.
The New York subway is the best way to get around the city and is an attraction in itself. Don't be afraid of the subway - trains are in much better condition than they used to be and people are well behaved. Buy a Metrocard from one of the machines (just follow the directions on the screen).
Exit the subway at the 59th Street station and walk west on 59th Street to 5th Avenue. The southeast corner of Central Park is now in front of you, and on 5th Avenue immediately to the south of you is some of the most renowned shopping in the world. Bergdorf Goodman (58th St and 5th Ave), Tiffany & Co. (57th St and 5th Ave), FAO Schwartz toy store (59th St and 5th Ave), the Apple glass cube (59th St and 5th Ave), and lots more.
When you've had your fill of shopping, either begin exploring the park by walking north through it, or return to the subway and go north to the 86th Street station and then walk east to Central Park and the Metropolitan Museum of Art. The Met is one of the world's leading museums and is filled with art that you're sure to recognize. Check the events listing for free tours available each day. The museum is gigantic so I recommend that you take one of the "highlights" guided tours. You'll save time and enjoy your visit more.
The Met Museum actually sits inside the edge of Central Park, so this is the perfect opportunity to explore. The winding paths through the park can be confusing, so it's a good idea to print a map of the park and bring it with you. Everyone has their own favorite sections of the park, but you may want to include Belvedere Castle, Strawberry Fields, Bethesda Terrace, Bow Bridge, and the Conservatory Water (remote control sailboats) on your tour. Riding a bike may be the best way to see the park, and you can rent a bike at the boathouse. In the summer you can enjoy Shakespeare in the Park or Summerstage concerts in Central Park.