Not surprisingly, since it is Oscar month, there are a number of excellent classic films and some "white elephants" (the latter I will not be recommending.)
April 1: George Cukor's intelligent comedy, Adam's Rib (1949) and Howard Hawks's great war film Air Force (1943)
April 2: Otto Preminger's Anatomy of a Murder (1959; John Steele Gordon discusses Robert Traver's novel in the March 2021 issue of "The New Criterion."); Leo McCarey's marvelous comedy The Awful Truth; Vincente Minnelli's lovely and melancholic The Band Wagon, 1953.
April 4: Chaplin's The Circus (1928) and Orson Welles's Citizen Kane (1941).
April 5: Frank Borzage's moving A Farewell to Arms (1932) and Minnelli's Father of the Bride (1950)
April 6: Truffaut's first film The 400 Blows (1959)
April 7: Fritz Lang's first American film, the corrosive Fury (1936)
April 9: King Vidor's early sound musical melodrama, with an Afro-American cast Hallelujah (1929)
April 11: Douglas Sirk's Imitation of Life (1959)
April 14: Mox Ophuls's elegant La Ronde (1950) and Preston Sturges's The Lady Eve (1941)
April 16: Leo McCarey's Love Affair (1939)
April 18: Anthony Mann's Western The Naked Spur (1953)
April 19: Ernst Lubitsch's Ninotchka (1939)
April 20: Roberto Rossellini's neo-realist Paisan (1946)
April 23: Nicholas Ray's Rebel Without a Cause (1955), Howard Hawks's Red River (1948) and Michael Powell's passionate The Red Shoes (1948)
April 25: Mark Sandrich's Shall We Dance (1937, one of the best Astaire/Rogers films), John Ford's Western She Wore a Yellow Ribbon (1949)
April 26: John Ford's Stagecoach (1939) and Alfred Hitchcock's Strangers on a Train (1951)
April 29: Lubitsch's To Be or Not To Be (1942, a very funny and very serious wartime film)