Oh dear. You're coming from SF, which is one of the great restaurant cities in the country. (No, they won't kick me out of NYC for saying that.
)
Thinking about uniquely NYC places -- first of all, I hesitate to recommend any place in Chinatown, just because.
However: you can't get more NYC than Katz's Delicatessen, on Houston Street. You want pastrami? You got pastrami. You want hard salami? You got it. etc.
And Russ and Daughters is just down the street, you can pick up some "appetizing" (cured and smoked fish and related) to take back to your hotel and/or bring back home.
Another experience is Peter Luger, possibly the most famous steakhouse in NY, and possibly the oldest. A nice subway ride over a bridge to Brooklyn, a short walk, and the best bacon and lamb chops anywhere. The steak is pretty good, too.
Bring lots and lots of cash, though -- no CCs, and it's not cheap. The wine list there is just eh, but there's a great new bar/lounge directly across the street.
OTOH, if you want prime rib, go to Old Homestead in the Meatpacking District. Lots of very trendy places there now (come in since I last worked there 6 years ago), now packing a different sort of meat.
It's been a while, but I used to like Macellaria. Mmmm, chicken livers over soft polenta.
For inexpensive and ethnic, you might head over to Ninth Avenue in the 40s and 50s -- Afghan, Brazilian, Argentinean, a pretty wide range, at reasonable prices. We especially like grilled fish at Uncle Nick's (Greek), and even though it's Chinese and you can probably get as good, Grand Sichuan is very well thought of. Both on the same block, I think 50-51st. Moderate prices at both.
For unusual, try Yakitori Toto on 55th Street, west of Eighth Ave. A Japanese place specializing in grilled foods. The chicken meatballs and tongue are especially good, and the only place I've had chicken sashimi. (Yup. Well, it's not totally raw.
). Good selection of sakes, too. Each item is not expensive, but it can add up.
If you want to go to one of Mario Batali's places, my first choice would be the pasta tasting menu at Babbo. Reserve now if you can. But: I also like Otto for antipasto items, the pizzas are nice, the pasta dishes can be good, and the gelati are sublime. Reasonable prices, too.
Another favorite place is Fatty Crab, in sort of the northwest corner of Greenwich Village. Malaysian-inflected, and a nice wine list. I love a place that smells of shrimp paste when I walk in.
The pork belly and watermelon salad is pretty close to a perfect dish to me.
Almost forgot: Craft. On the expensive side, and very easy to over-order because everything is a la carte. But oh! the vegetable sides!! Actually, if your wife can cajole a bunch of people into joining you, so much the better. It's really good when you have more people, so you can share.
I haven't been, but many of my friends say that Sripaphai, in Queens, is THE BEST Thai restaurant they've ever been to, even in Thailand! Very reasonable prices.
Well, that's just a few. You might want to have a look at the
New York board on Mouthfuls, where people discuss places all over the city. Chowhound, of course, also has extensive discussions.
And if you need things to do, check out Chelsea Market and the Union Square Greenmarket (USGM is not open on Thursday). There's a thread somewhere about a bunch of us who got together a few years ago and hit both of them, plus Craftbar (which has since moved, but is still good). In nonfood stuff, I recommend one of my nearby museums, the National Museum of the American Indian, down by the Battery at the Custom House; you actually get two museums for the price of one, since the building is really neat, too -- and the price is free!
Walking around the South Street Seaport Museum is also good, if the weather is nice. Skip Pier 17, just a big mall.