I always hit the art museums and the American Museum of Natural History. Maybe a Broadway show, if any of them look good. You might also check for sports events and concerts on the days you'll be there.
My restaurant list would include something world class that I couldn't get elsewhere. I'd try to get into one or more of these:
Le Bernardin
Per Se
Babbo
All of these will take serious money and you want to get a reservation now, if possible. Figure that you won't get a table anywhere decent if you wait until you get there to make the reservation.
There's a million other places to eat. I'd just research it online. Avoid the lists compiled by bloggers, food critics and the NY times. Stick with the list from Michelin, Zagat, etc. as they will be more consistent and not favor the "newest is best' sort of mindset. A 5 second search turned this up:
Zagat, Michelin Best Restaurants In New York City - Business Insider
There are others...
JR