Here are some movies I liked a lot:
La grande bellezza (same director as Il Divo. A movie about the emptiness of the modern life)
La dolce vita (La grande bellezza is basically a modern La Dolce Vita. Same subject, different generation)
Ladri di biciclette (hyper realistic movie from the 40's with a story more relevant than ever)
Malèna (if you haven't already, you will fall in love with Monica Bellucci)
Amarcord (small, funny tales from an Italian town)
L'uomo delle stelle (a man claiming to work as a caster travels around in Sicily and steal people's money)
I cento passi (The story about a guy in Sicily who stood up against the mafia)
Il gattopardo (costume drama about a time long ago and how "a leopard can't change its spots")
I'm not counting Sergio Leone's movies, but if you want to, then watch them all. Masterpieces.
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Malèna, L'uomo delle stelle and I cento passi are not masterpieces, but easily accessible.