Tons of fun for teens in the city, including:
Canal Street doesn’t disappoint. Stroll around the area to get your bearings. Walk up and down Canal and a few of the cross streets, poke around in the shops, buy some souvenirs and accessories, then get down to the secret agent mission of knock offs.
The shops along Broadway between Canal and the NYU area are great.   A few shops to note on Broadway: H&M is here, Le Chateau has great service and there are many other chain stores and boutiques throughout this area.

Great street vendor shopping where you enter/exit the Ellis Island/Statue of Liberty cruise. 

See the sights:
Empire State Building - If you buy the combo ticket online it includes the Sky ride and Observation Deck and you DO NOT need to wait in the main line. Enter the building at the 33rd street entrance and, turn left at the corridor. Go up the escalator at the very end of the hallway, (by the pizza place).
Tussauds Wax Museum is really fun. Get your picture taken with Jerry Garcia, Brad Pitt or any number of waxed celebrities.
Ellis Island is great for families.   It's very well laid out and has interesting museum. There are great photo ops of Manhattan from the big arched windows and the ferry ride is fun too.
Ride the old, wooden escalators at Macy's .
Brooklyn Bridge.   Take the train to High St. in Brooklyn and walk back to Manhattan. Including stops for pictures and to admire the view, it takes about an hour.
Rockefeller Centre observation deck has a better view than the Empire State Building.  If you have time, do both!
New York Public Library and the Grand Central Station .

Neighbourhood walk through Chinatown, Little Italy, SOHO and Greenwich Village .

See a show:

Mamma Mia
The Nutcracker
Blue Man Group

Tavern on the Green is only worth it for the atmosphere, especially the Crystal Room. Otherwise, the food is not good and overpriced for what you get. It resembles a big tacky wedding. Go for a drink and a photo op.
Carnegie Deli is great for lunch and the sandwiches are outrageously huge, so two sandwiches and an order of fries is more than enough for 4 people. Save room for cheesecake.
Grays Papaya for great hot dogs.
Serendipity 3 - very quaint restaurant.  Try the frozen chocolate dessert!
Ellen's Stardust - singing waiters are a treat!.  Great location.
As a general rule the law in New York forbids the presence children under the age of 16 in bars unless their parents accompany them.   The individual establishment may have rules of its own, and you would be wise to check ahead of time.   Sometimes, bars are more restrictive if they do not serve food.

** Note:  Some of the original information for this thread came from the following forum thread: -