Italian...
Lil Franike's: 19 1st Avenue, New York, NY 10003
Steak...
Keens: 72 W 36th St, New York, NY 10018
Peter Luger:. 178 Broadway, Brooklyn, NY 11211
For Italian... Beware of Little Italy. Most of the places there are terrible Italian food. Tourist trap restaurants. Sub par food with high prices. It is a fun place to walk around though and a "used to be hidden" speakeasy with relaxing outdoor seating in the back is right in the middle of Little Italy. It's called the Mulberry Project at 149 Mulberry St, New York, NY 10013. Bespoke cocktails made fresh. Nice place to sit outside in the back if weather permits. No sign. Enter through an unmarked basement door and impress the wife with your knowledge of secret NY. :)