Film noir:
The Postman Always Rings Twice (the John Garfield/Lana Turner version, NOT the Jack Nicholson/Jessica Lange version)
Nightmare Alley
Double Indemnity
The Maltese Falcon
Sunset Boulevard
The Dark Corner
Out of the Past
Criss Cross
The Killers (the Robert Siodmak and Don Siegel versions are equally good, get either)
Leave Her to Heaven (one of the few color movies labelled as noir)
Angel Face
Gilda
Night of the Hunter
The Big Clock
Italian movies:
Ossessione (Italian version of The Postman Always Rings Twice, very interesting to see how a different director handles the subject matter)
1900 (epic historical movie by Bernardo Bertolucci)
The Leopard (another historical drama based on a very famous Italian novel, Il Gattopardo, directed by Luchino Visconti - stars Burt Lancaster in the title role)
The White Sheik (an early Federico Fellini film - a very light, comic film, my personal favorite of his)
The Conformist (another Bertolucci film, an absolute knockout, go into it blind without knowing anything about it - starring Jean-Paul Trintignant)
The Legend of 1900 (a personal favorite of mine, and a bit of a guilty pleasure - not a great film, but the soundtrack is dazzling. Directed by Giuseppe Tornatore, who also did Cinema Paradiso)