NYC - my hometown! I'll assume you're staying in Manhattan. Mass transit's excellent for the most part, so you can get around via subway or bus.
Political Comment: KOCH FOR MAYORAL RE-ELECTION IN 2009! If Bloomberg can bypass the citizen's wishes for term limits, then former Mayor Edward Koch should take advantage of the situation. He knows financial crisis and has a decent sense of humor, unlike GLOOMberg.
Anyway, back on topic: Check out Chinatown in downtown Manhattan. 17 Mott Street is my favorite - Wo Hop Restaurant - but use the BASEMENT restaurant. The kitchen's better. Try the BBQ spare ribs - they're heavenly. $$$ are reasonable. Little Italy's just a few blocks north - pick up some cannoli's and have a capuchino. If you want italian food, there are some good restaurants left. Check out Froders for suggestions; I haven't eaten there in years.
You'll want to see City Hall and the Woolworth Building in downtown. The former site of the World Trade Center is just a few blocks west of there. If you want to shop for Electronics, you can't go wrong with J&R Music World on Park Row across from City Hall. Wall Street is a few blocks south. If you head southeast, you can get to the South Street Seaport which has a small mall on the East River. They usually have street performers and the bars are hopping at night.
Head for SoHo (South of Houston Street) -- there are tons of trendy restaurants. BTW, "Houston" is pronounced "HOW-stun" not "HEWs-ton" like the city in Texas. There are Clubs here and there throughout the area up into the numbered streets.
Don't miss Greenwich Village - it's eye-popping, esp. at night. Their annual Village Halloween Parade is coming up this month and it's a blast to see/watch. Like Mardi Gras with drag queens and political statements, but all set to music and lots of fun. Everyone should see this at least once in their life.
There used to be a small chain restaurant called "America" in the 20's near the Washington Square Arch. They serve all sorts of american food, including Fluffernutters. (Peanut butter and marshmallow spread sandwich; traditional US junk food. As popular to Americans as vegemite is to Aussies.)
Midtown - On the East side of 34th street (the avenue numbers go from east to west), you'll find the NY Daily News building. That was used for the Superman movies. Walk west on 34th street and you'll find the Empire State Building at Fifth Avenue.
From the Empire State Building, the famous Macy's Herald Square store is due east on Broadway and 6th Avenue. (AKA: Avenue of the Americas. AOTA Don't ask why 6th avenue has two names. People will know you're a tourist. NY'ers dont' know and don't really care, lol.)
Before you head uptown, please take my advice. Do not (repeat: DO NOT) play "three card monty" with anyone. Even if you win, their shill will mug you once you turn a corner. Also, all the stuff you see for sale is FAKE - no one sells Gucci and Prada on the street. It's like some strange bazaar. You can bargain for some things, esp. towards the end of a day.
As for the food carts, I've heard the Taco trucks are okay, but stay away from the Souvlaki ones. Hot dogs are hot dogs - look for a Sabrett umbrella/sticker and you'll be okay. The pretzels are good if they're fresh, awful if they're stale and dried out.
If you head up Avenue of the Americas, you'll reach Bryant Park at 42nd Street. They have a free ice skating park in the winter, but it'll be packed up for Fashion Week before you arrive in March. Admire the Grace Building's very cool curves as you walk east to 5th avenue alongside the park. (There are chess players, but I think they charge $5/$10 to play a game against you.) Around the corner, you'll see the famous NY Public Library building - set of many movies, including one of my favs: Ghost Busters! The lions have names. If you have time, take a tour - the library's HUGE and has a lot of history. (I dare you to ask where the slimed card catalogs are from GhostBusters, lol. I think that was a set, but I'll bet the tour guides get asked it regularly.)
Further north, you'll find really interesting office buildings for major corporations. Most have some kind of art gallery in the lobby, to justify their existence. They also have "greenspaces" or small sitting parks required by law because they blitzed every bit of green in midtown to build those buildings.
When you get to 50th Street, Radio City Music Hall will be on your right. Their backstage tours are great, as is the NBC Studio Tour in the NBC building behind it. Walk to the east and you'll find the most famous ice rink in the world - Rockefeller Center, where the giant Christmas Tree is lit each year. (I would NEVER go to that again. Not an inch of personal space and you can see better on TV)
EDIT: NBC's "Today" show often uses Rockefeller Center as their outdoor set, and crowds are always welcome. They have skating exhibitions or special guests on their outdoor stage. I think they have a weekly concert too. Same deal with the ABC's "Good Morning America" broadcast - they broadcast crowds and concerts from Times Square. Check out their website if you're an early riser. (Get there before 6:30am) Get a chunk of poster board, make a fuss about your pilgrimage and give us a shoutout if you get on camera!
When you arrive, the rink will probably be undergoing the annual transformation for the NY Flower Show. If there's ice skating, the cafe downstairs has a pricey watch-and-eat location. On street level, there are all sorts of high-end designer stores and jewelry shops. Don't buy jewelry there - go to the Diamond District on 47th Street between 5th & 6th. If you don't know what you're buying, caveat emptor. BTW, don't buy jewelry from sidewalk stands and the super-cheap videos are often poor quality bootlegs. My ex-SIL once gave my daughter a Cinderella tape that had the end of a porn flick at the end after Cinderella was over.
Enter through one of the buildings on 6th avenue and follow the signs for the underground Concourse. It's an underground passageway that leads to subways and tons of restaurants and shops. It runs from 47th street to 52nd street, all underground. Great place to go if it's raining.
Turn west on 54th Street and you'll find the Ziegfield Theater, which used to be a vaudeville showcase venue. Lots of movies have their premieres here; sometimes if you ask nicely you can find out in advance or see early showings before the movie's released. It's beautiful inside, with art deco architecture/decorations.
On 7th avenue near 56th street, you'll find Carnegie Hall without even practicing! Cross over and order a Pastrami or Corned Beef sandwich at the Carnegie Deli - it's incredible! (Get a cream soda with it, please. Don't ask for Mayonnaise-they keep kosher; just use mustard.) A little pricey, but if you get the 5" high sandwich "to go" and split it with someone, it's perfect for lunch. Oh, and the rule is, if one person in a group eats a garlic pickle, EVERYONE has to eat a garlic pickle...just sayin' lol.
Walk uptown (which means "north") and you'll reach Central Park. They have hansom carriage rides with horses, which is nice. The park itself is very big - get a map. There's a large outdoor skating rink (winter=ice, summer=roller) and the bridge from the snowball fight in "Elf" is nearby.
The east and west sides of the park's perimeter streets are lined with apartment buildings and some nice private homes.
John Lennon/Yoko Ono lived on the West Side in the Dakota building, which is where he was shot. Strawberry Fields is the memorial garden inside the park dedicated to his memory - it's not far from his building.
The Museum of Natural History is waaay uptown at 79th street, so I would take a cab if you're interested. Things have changed since I was young, but many of the things from the "Night at the Museum" movie are still around. There are 23 buildings in the museum and the movie never revealed the Space part of the museum with the famous meteorites, moon rocks and "what you weigh on (planet/moon/sun/etc.)" scales. (Sounds like a sequel, doesn't it?) Great place, allow at least an entire day to see it all and prepare to be exhausted.
On the East side of Central Park, the giant balloons for the Macy's Thanksgiving Parade (Novembers) are inflated the NIGHT BEFORE the holiday. It's a fun event, very family-oriented. Sorry you wont' be able to see that during your visit.
Heading down fifth Avenue to 59th Street, you'll find Madeline's Plaza Hotel, which is now mostly condos but they still have the tea salon on the ground floor. Keep going and you'll find all the famous Fifth Avenue stores, including Tiffany's, Saks, and Lord & Taylor. The American Girl and Disney stores are pretty cool, and you can get lost in the Apple store for certain, lol.