Back in the early '70s, there were maybe 10 repertory theatres all over the greater Los Angeles area showing different movies every night of the week. Most of them are gone now, killed by the home VCR and cable TV. (How ironic: The same devices that have helped me make my living since 1997.) But there are several in the Los Angeles area carrying on, and I make it a point to check them out every week to see if there's something good worth seeing again, or checking out.
I recommend these, and present them in alphabetical order:
The Academy of Motion Picture Arts and Sciences has ongoing film series, events and lectures. The Samuel Goldwyn Theater is one of the showplace movie theatres in the Los Angeles area, and the Linwood Dunn in Hollywood is up to that standard.
The shows are a bargain-basement $5, although they often sell out.
American Cinematheque operates two theatres in the Los Angeles area, Grauman's Egyptian in Hollywood and the Aero in Santa Monica. They often show the same films at both houses, so check the one close to you first.
The Los Angeles County Museum of Art has an ongoing film program, usually focusing on retrospectives of actors, directors, or themes. Some of my best filmgoing memories come from the Bing Theater, including the night Alfred Hitchcock stepped into the audience and watched "Rear Window" with us, watching David O. Selznick's personal print of "Gone With The Wind" (a 1.33 three-strip Technicolor nitrate print that was not faded from overuse), and the night somebody had the effrontery to ask Howard Hawks who the real director of "The Thing From Another World" was. (Hawks sidestepped the question.) http://lacma.org/programs/FilmSeriesSchedule.aspx
The New Beverly Cinema is L.A.'s premier rep house. The recent death of owner/operator Sherman Torgan was a terrible blow, but the place was bought by Quentin Tarantino and should go on for decades to come.
UCLA has an active film program, although the screenings are held off-campus at the Billy Wilder Theater in the Occidental Petroleum building at Wilshire and Westwood. I miss the old screening room at Melnitz Hall on-campus.
Finally, if you must go to a commercial theatre, check out the offerings at Laemmle. Founded by Max Laemmle, a nephew of Universal founder Carl Laemmle, they have been in the forefront of presenting foreign and independent films in Los Angeles since the '30s. If memory serves, they showed François Truffaut's "Shoot the Piano Player" at the Royal for a couple of years after it came out in 1962. I do know this: On their website they offer a T-shirt that says "Not afraid of subtitles."