The Manhattan Little Italy has dwindled to a mere cipher of what it once was, due to expansion from neighboring Chinatown, rising rents, etc. So much of what is left is really for the tourists, and the San Gennaro and San Antonio festivals.
New Yorkers tend to look at Arthur Avenue in the Bronx as the "real" authentic Little Italy. Plenty of Italians and Italian Americans have kept the numerous shops, restaurants, in this area right to the south of Fordham University. There are actually "trolley" tours that take in the Arthur Avenue Italian neighborhood, along with the New York Botanical Gardens, and the Bronx Zoo (which are nearby).
Other than that, take the Metro North commuter train to Fordham station, step out onto Fordham Road, and it is a fifteen minute walk to the Arthur Avenue Italian neighborhood. The heart of it is at Arthur Avenue and 187th Street. You will find tons of homemade cheese, fresh pasta, butcher shops, pastry shops, etc. You really can’t go wrong at any of them.
If you have interest in eating at restaurants there, let me know, I have my personal favorites.
You can even google the Arthur Avenue Italian Market…they have a nice website that gives history of the neighborhood, etc. Mike Greco, who runs it (and owns the great deli there) is a fountain of great info on the neighborhood.