I would stay away from the celebrity chef places, way overpriced in most cases and not worth it. Now, if we are talking Italian food.........head down to Little Italy and look for a place called Peasant. I have no idea exactly where it is but google it. Probably the best authentic food I have had since being in Italy itself.
Ok, if you want to experience the downtown aspect with some Mexican.......go to Chevy's on W42nd. Great place to people watch and they make their own tortillas in the middle of the place. I forgot to mention a huge full-bar with many tequilas. My first trip to NY, the wife dragged me there and we met Mariah Carey.