Definitely see a Broadway show. Whoever suggested In The Heights is right on the money, it's an excellent show for teenagers. I would also suggest Wicked, South Pacific or Bye Bye Birdie (it's opening in September, starring John Stamos as Albert).
Shopping-wise: I love SoHo. There's so many cute boutiques around the Broadway/Prince St. areas. Also, Canal St. btw Broadway and Mulbery St. has some awesome stores, and if you ask the right question to the shop owners, you can buy really good quality knock-off designer purses. I bought a decent sized Coach purse for $27, and you can't even tell it's a knock-off.

There's also the South Street Seaport, I highly recommend there too.
Other fun things: Take a walk across the Brooklyn Bridge. You get awesome views of the city/Brooklyn, and better yet, it's free!
Statue of Liberty/Ellis Island. I'm slightly bias as I'm interning at Ellis right now, but I love it here. The Park Rangers are so knowledgable about the subject matter, and love sharing their knowledge with visitors. And looking up your heritage, seeing the names on either the wall or on the database is so cool.
As for the Statue, I highly recommend trying to get passes to go up to the crown. It's a big of a hike, 300-someodd steps in an enclosed spiral stairwell, but it's worth it. If you don't want to go all the way up, get monument passes to go up to the pedestal. Get both of these passes well in advance, especially the crown. It's already sold out until mid-September.