Pre-1960s cinema is awesome! Some of my personal favorites are:
The movies of Ernst Lubitsch:
Trouble in Paradise (usually considered his masterpiece, 1931).
Design for Living (Gary Cooper, Miriam Hopkins, and Frederic March agree to have an MFM menage á trois relationship. Things go... Lubitschy. 1932)
Ninotchka (haven't seen this yet, but it's one of Greta Garbo's most iconic roles, 1939)
To Be or Not to Be (Perhaps best remembered as Carol Lombard's last film before her tragic death, Lubitsch elegantly tells the story of a group of Polish stage actors and their plot to survive in Nazi-occupied Warsaw. My personal favorite of his, 1942).
Heaven Can Wait (After he dies, a guy's only ticket out of Hell is more or less to give the devil a play-by-play of his entire sexual history. I found this much more endearing the second time, 1943).
Marcel Pagnol's "Fanny" Trilogy: Marius, Fanny, and Cesár (1931-36). The trilogy tells the epic love story between a taverna owner's son and his cockle shell-selling lifelong friend in the French port city Marseilles. There is much to say about these films and the beauty within them. Thanks to the criminally forgotten character-actor Raimu, Marius, I believe, contains both the most realistic and poetically beautiful depiction of paternal love ever brought to the screen. And Fanny, in spite of its simple premise, is so skillfully told that it's probably the film to have ever put me on the edge of my seat in utter suspense--without anybody getting killed! Cesár, while not accomplishing the dramatic heights of the previous two, is nonetheless a fine film, and a fitting good-bye to a world many have come to love.
I Remember Mama (1944): What The Fanny Trilogy was to fathers and sons, Mama is to mothers and daughters. While there is no general plot, the film is mainly a series of vignettes illustrating the domestic life of a family of Norwegian immigrants in 1910, particularly how far Mrs. Hanson (Irene Dunne) is willing to bend to the will of her maternal instinct. A charming and touching film indeed.
The films of Powell and Pressburger: If you like Kubrick, I say go for Black Narcissus. If you like pleasanter fare, I suggest the romance I Know Where I'm Going!, the romance-fantasy A Matter of Life and Death. More importantly, The Life and Death of Colonel Blimp --their magnum opus-- is probably cinema's best treatment of masculine camaraderie.
Billy Wilder is good, and a disciple of Lubitsch. I recommend seeing Sunset Boulevard and Sabrina, which are both on Netflix I think.