Local Museums and Cultural Institutes:
American Film Institute - this is Hollywood, after all!
Getty Center and J. Paul Getty Museum - see esp. their "Painting on Light" exhibition (July 11 - Sept. 24)
Huntington Library, Art collection, and Botanical Gardens - out in Pasadena, but definitely worth the trip!
Los Angeles County Museum of Art (LACMA) - see esp. "Gold of the Nomads: Scythian Treasures from Ancient Ukraine" and "Portraits of Power, Portrayals of Piety"
Museum of Contemporary Art - downtown
Museum of Tolerance - focusing on the Holocaust
Norton Simon Museum - more fabulous art, if you're going to Pasadena
Skirball Cultural Center - see esp. their exhibition on "Sigmund Freud: Conflict and Culture"
Entertainment: