I used to visit NYC when my wife studied there a few years back. The places I liked were usually more low-key and not fancy (especially since there's not much catering to vegetarians), but serve good food to be shared in good company.
http://www.yelp.com/biz/minca-new-york
Best ramen I've had outside of Asia, and the salads aren't bad either.
http://www.yelp.com/biz/chikalicious-dessert-bar-new-york-3
Novel and fun dessert bar with a prix fixe menu
On the other hand, the wife is a foodie and non-vegetarian, so she recommends these places (if they are still around):
Katz Deli: Famous pastrami sandwich, featured in When Harry Met Sally
E.U.
Blue Ribbon Sushi
Stanton Social
Otto
Tia Pol
Schiller's
Casa Mono
Republic
Shake Shack
Burger Joint
Yakitori Totto
Cookshop
Hearth
Sapporo
Also check out the "Best of NY" by NYMag for other suggestions:
http://nymag.com/bestofny/
Hope you have a great vacation!