If you want to get the old neighborhood feel, I'd suggest going to an old school restaurant in the outer boroughs. I'd suggest Bamontes or Marco Polo in Brooklyn or Park Side or Don Peppes in Queens. The Brooklyn ones I suggested are very close to Manhattan. Little Italy in the Bronx has some good ones too but it will be kind of far for you.

Definitely go see the other stuff, but after you go see the ravenite is a bougie women's shoe store etc... I think you will want something with that old feel. Also I would avoid most restaurants in Manhattans little Italy. Most are overpriced and not great.