1st Time in Manhattan
-- Grand Central Oyster Bar...sit at the counter and, assuming you like them, order only oysters (both raw and fried) plus maybe some salad or coleslaw...they also have a huge wine by the glass list...a classic NYC experience...
-- one of the pizza places mentioned above...i'm partial to Grimaldi's, but John's is good too...personally i've had substandard meals at Lombardi's and won't return...
-- get a bagel and smoked salmon to go (it's a store not a restaurant) from Russ&Daughters on E. Houston St...amazing amazing smoked salmon: try a couple different samples and have them make you one to go...(and if you like pastrami, Katz's another classic is 50 ft east and you can eat your bagel&lox there alongside your pastrami)...Russ&Daughters is easily one of my top three food places in the city...
-- an old-school Italian place...i like Il Giglio (upscale but not crazy expensive as long as you avoid dishes laced w/ truffles) and Piccolo Angolo (moderate)...and a search here will net you dozens of other similar Italian suggestions...avoid any restaurant located in Little Italy...
-- Lupa...Mario Batali's Roman trattoria...you can get a bunch of antipasti (brussels sprouts, the beets w/ pistachio, and the octopus w/ chick peas are my three favs) plus a pasta or some smoked ham...or try his slightly less expensive place Otto...
-- Chinatown...Great NY Noodletown for bbq pork and salt&pepper squid and fish w/ flowering chives...dimsum at a variety of places but lately i go to 88 Palace at 88 E.Broadway, but Jing Fong and others lauded here too...or a whole fish somewhere like Ping's or Fuleen...
-- and, while it's not a must, n33 (w/ a Spanish accent on the n: "en-yay") at 33 Crosby is a tiny, trendy little tapas/wine bar that you might enjoy...
Wow. What great responses!
I agree that you definitely have to have pizza while you're here.
For first timers, Grimaldi's would be my suggestion, even though all of the places mentioned here are great.
Grimaldi's is right by the Brooklyn Bridge, and -- weather permitting -- after you eat your pie, you can walk across the bridge and take in the awesome views that it gives you of the Manhattan skyline. It's really an experience.
I'd eat by category...hitting one of each of the NYC classics (and adjusting for neighborhood):
Bagel Shop: Ess-A-Bagel (various locations)
Deli (Pastrami): Katz' (Lower East Side) Sarge's (Midtown)
Pizza Pie: Arturo's (Village) or Naples 45 (Midtown)
Street Cart (Lamb Gyro): 44th and 6th; 28th and Park
French/Fancy: Artisinal (wine and cheese especially), Raoul's (scene)
Italian Bakery: Rocco's
Italian Sub: Faicco's (cold); Lazzara's (hot..sausage and peppers, chick parm, meatball)
Red Sauce Italian: Volare
Dim Sum: Pings (Chinatown); Chiam (Midtown)
Non-NYC specific categories:
Burger: Shake Shack; Chumley's; Westville
Brunch: Jane; Norma's
Quaint Italian (in West Village): Valdino West; Da Andrea
There are so many great restaurants here I don't know where one could ever begin. If you like wine and plan to wander around shopping in SoHo, I'm heartily recommending Centovini, on Houston b/w Mercer and Greene a couple of blocks west of Broadway. It's not cheap, but it combines good food, good wine, and great atmosphere. Plus the staff's great.
ALSO... it's Brooklyn restaurant week. Look it up online here (http://www.visitbrooklyn.org/dining_Dine_in_Brooklyn_2007.html). Lots of great restaurants for under $30 prix-fix if you're willing to take the subway.
Second to all the above pizza places. Add to that Arturo's pizza on Houston. There is a TON of debate as to whether or not they are better than John's. But Arturo's has live jazz!
Stay away from any pizza place that loudly claims to be "Famous" or "Original" or named after some guy called Ray. These places may have better pizza than most other places in the entire USA but they are usually the worst in NYC.