As somebody who lives in NYC and eats out 3 times a week it's difficult to offer recommendations concisely. I'll just pick a few standouts
I think The Red Cat is one of the best tables in the city right now. If you take the advice of walking the Highline it would be a great idea to have lunch or an early dinner there. The Red Cat is on 10th Ave between 23rd & 24th.
Seasonal Restaurant & Weinbar is an Austrian restaurant with a Michelin Star, casual dresscode, and $27 three-course lunch. It's on 58th between 6th Ave & 7th Ave so ut makes a nice stop between Central Park and the shops on Madison Ave.
If you're into offal, Yakitori Totto is a great option if you're in the Columbus Circle area.
I'll be the third to recommend Ippudo, but add that in addition to being the best ramen place in NYC they also have the best steamed pork buns and some of the best wings in the city. I wouldn't leave without ordering all three.
Graffiti in the East Village is a very interesting and well-executed spin on Indian haute cuisine. It's a very small place with communal seating, but the service & food are worth it.
It's not a restaurant, but a tour at the Lower East Side Tenement Musem is something I suggest for any first time visit. The building is only open to people taking tours, so I suggest booking in advance through http://www.tenement.org/
There are a lot of interesting things to do in the surrounding area (Chinatown is just South, SoHo is just East), and the Lower East Side is one of the restaurant neighborhoods in the city.
No matter what you do I hope you enjoy your time in NYC!