After having trashed Little Italy and Times Square, I should have at least offered alternatives.
Yes, there are many very good Italian restaurants are all over the city and you will be overwhelmed. If you want casual dining near at or within walking distance of Little Italy and GV, you might want to check these out:
Peasant in NOLITA: can be a little noisy, but always fun for a small group, food and decor are rustic, others say romantic, very good regional pasta dishes, sufficiently robust Italian wine list, under-rated Neapolitan pizza (the margherita is always a must for me), and the roast crispy pork dish rivals any in the city and, imho, including the Mailaino version which is already very good. http://peasantnyc.com/
Rubirosa in Nolita: Good for families. Solid food and OK wine list. Casual. Very good service. http://www.rubirosanyc.com/
Babbo in GV: one of the best in the city at regional Italian cooking. Plenty of A- to A+ dishes to choose from their antipasti, primi, and secondi menu. Very big Italian wine list, at all price points. Tough to reserve. Noisy and bustling. Rock music. Even though I don’t care much for this, but I’ll just mention it - Mario Batttali. http://www.babbonyc.com/ristorantemenu/ristorante/
Malatesta in GV: Serves some of the simplest yet tastiest Italian trattoria dishes. Very casual. Small wine list. Wine glasses that are of the small Italian non-stemmed tumblers. Low to moderate costs. Mostly young Italian-speaking wait-staff. Romantic, if you are looking to eat in a setting similar to a restaurant overlooking the piazza in a small Italian town. http://www.malatestatrattoria.com/
Keste in the GV: Mostly pizza. Thicker than your basic Neapolitan. Well-made delicious dough and toppings that’s as good as theyr’e done locally. Just for the record, this still not my local Manahttan-only favorite pizza place. Home – kesté - Nationwide Shipping
Da Marcella in GV – Romantic and dark (may be too dark for others). I like the cool , yet family vibe in this restaurant. Gems in the daily specials. http://www.damarcella.com/Home.html