Wow. I'm just getting around to seeing this now. Very sorry. I'll have to check in here more often. Although I am back to more regular participation on these forums, because of time constraints, and stress due to extreme financial difficulties, I've confined my reading and posting only to certain forums.
One thing I'd have to say before getting started is that I'm very unqualified to answer this post properly. I don't go out much, and I never have. As a result, there are a lot of things that go right by me, when it comes to cultural aspects of NYC. I don't like going out much, and being on the verge of losing my apartment, and perhaps a lot more, I don't eat out at all. I never did eat out much, although I did have my favorite spots, but these days, all eating out, even stopping for pizza somewhere, is out of the question.
If you're in Manhattan, I'd say check out Imperial Pizza on 34th and 3rd Avenue. In Astoria, Queens, there's Grand Pizza on 35th St. and 30th Avenue, or Polito's Pizza on Broadway near Steinway Street. As for Jewish Deli's, there's a great place in Forest Hills, just outside the 67th Avenue subway, called Knish Nosh. In Manhattan, there's Katz's Deli on Houston Street, way downtown.
As for good Italian Deli, I'm really unqualified for that. Terribly sorry, but I hope I've helped a bit.