The above suggestions are all good, but if you are looking for red-sauce Italian-American I would add Rubirosa. It is on Mulberry St., just a couple of blocks north of Little Italy, and the food is far superior to the Little Italy places. For something more sophisticated, try Ciano (listed above in kathryn's post) on east 22nd st., another relative newcomer that is turning out some very serious food. We were there last night for the second time, and it's really exceptional.
Other upscale Italian options:
I haven't been to Il Mulino, Il Matto, Casa Lever, Porsena, and Novita, but these are worth exploring as well.
86 W 3rd St, New York, NY 10012
243 E 58th St, New York, NY 10022
102 E 22nd St, New York, NY 10010
390 Park Avenue, New York, NY 10022
281 Church St, New York, NY 10013
21 E 7th St, New York, NY 10003
I'd stay out of Little Italy personally. There are other places to go if you want red sauce Italian-American.
Italian is very hot in NYC right now. I would start researching:
Marea - especially for seafood
Del Posto - especially their weekday prix fixe lunch deal
Esca - especially for seafood and their crudo
And for less expensive (but not exactly cheap)
'ino and 'inoteca
There's probably more I'm forgetting!
Babbo vs Scarpetta
Best Italian for my NY Foodie trip
Where does a NY visitor find THE BEST Italian food?
Ex-NY Hound visiting from LA - help me get my italian fix!
Upscale Italian for lunch/dinner: Marea/Scarpetta/Convivio?
Affordable Rustic Italian near Greenwich Village
Italian: I've narrowed it down to Five...
170 Thompson Street, New York, NY 10012
110 Waverly Pl, New York, NY 10011
85 10th Avenue, New York, NY 10011
240 W 14th St, New York, NY 10011
68 Clinton St, New York, NY 10002
355 West 14th Street, New York, NY 10014
45 Tudor City Place, New York, NY 10017
228 West 10th Street, New York, NY 10014
240 Central Park South, New York, NY 10019
377 Greenwich St, New York, NY 10013