I live in NY. Don't stay near Times Square/midtown. I suggest the Flower District/Chelsea/West Village.
For some higher end dining teens would like I'd suggest:
Tao Downtown
STK Meatpacking
The lunch special at Milos Hudson Yards (amazing views!)
Empire Diner, which is in an old 50s diner, but now a v.v.v. trendy restaurant. Gallow Green is the restaurant in the same building as Sleep No More and is very cool and hidden away.
All the branches of Boucherie are very instagrammable.
RH Rooftop on top of Restoration Hardware
Legacy Records (in an old recording studio)
For cheap food I like:
Los Tacos no1
Fuku at Hudson Yards,
Shake Shack (always)
Beatnic for vegan, Honeybrains and Sweetgreen for salads/rice bowls on the go Daily Provisions for yummy breakfasts and the best BECs ever
Eataly is lots of fun and they have cheap and pricy options.
Xi'an Famous Foods for authentic, fast casual Chinese food
Brooklyn Bagel for the most gigantic, softest bagels. Get them fresh from the oven!
Van Leeuwen ice cream
I recommend walking the full length of the Highline right down to the Whitney Museum and then checking out the exhibits there. You can also go to the new park - Little Island - which is built out in the river. They usually have some fun free stuff going on too.
Gantry Plaza State Park in Long Island city is a gem - IMHO some of the best skyline views you'll see. You can also go to Casa Enrique which has a Michelin star and won't break the bank. Also worth checking out some batshit contemporary art at MoMa PS1. If you're going in the summer, check out their Summer Fridays live DJ sessions in the late afternoon.
I loved Museum of Ice Cream and the Color Factory. They're 'instagram museums" but really fun. Similarly Artechouse under Chelsea Market is a great immersive arts experience.
Things to skip - Ellen's Stardust Diner (ripoff tourist trap), buying anything in, or near Times Square, M&M world, pedicab rides, horse drawn carriage rides, Trump Tower.