- The ferry to Staton Island is free, and goes by the Statue of Liberty and gives you a nice view of Manhattan.
- Walk in Central Park.
- Check out Union Square: there's a green market, shops nearby that teens will like, and great people watching (chess players, acrobats, dancers, singers, and everyday and not so everyday people).
- A quick stop in Times Square.
- Soho if they like shopping (start along Broadway).
- A museum like the Met or MOMA.
- A Broadway show.
- Soul food in Harlem.
- Make sure you ride the subway somewhere.
Just walking around Manhattan is great - you get a real sense of the city and its people.
My list of recommendations, with must-sees highlighted, is at http://traveleater.wordpress.com/nyc-field-notes/ over 5 years ago