The Best Touristy
In mid-town near Times Sq- Junior's (cheesecake), Sardi's, 21 Club , Keen's,.
None of the first 3 are standouts for food. Keen's, on the other hand, is one of the best steakhouses in ther city. Katz does get lots of tourists, but plenty of locals- excellent pastrami. Babbo in the West Village is another restaurant very popular with tourists (and the hardest reservation to get). Gray's Papaya is another NY institution, go for the hot dogs.
110 Waverly Pl, New York, NY 10011
21 West 52nd Street, New York, NY 10019
539 8th Ave, New York, NY 10018
The Oak Room at the Plaza is the only spot that comes to mind now. I've no idea the quality of the food there.
[Great question, petunia; I look forward to seeing what the c'hounds drop at your feet.]
EDIT: I notice the auto-google-map inserts the Oak Room at the Algonquin Hotel. That's really a cabaret, like Feinstein's; not actually a foodie place and not the one I meant at first. Still, the restaurant in the back of the Algonquin lobby, where the Roundtable met, is pretty good for food, and it's a very nice room. The only reason you may not want to go there is that it feels as though the lobby lounge and that dining room in back are actually filled with New Yorkers, and not with tourists, apart from the hotel guests.
59 W 44th St, New York, NY 10036