Variety is the leading trade paper of the U.S. entertainment industries, including the film industry. You should work closely with it for your assignment, though there are many other trade papers that you can use as well. Coverage of Variety varies greatly depeneding on where you are accessing it--see below.
Entertainment Industry Magazine Archive - The Entertainment Industry Magazine Archive contains full runs of key trade and consumer magazines in film, broadcasting, popular music and theater, including Variety and Billboard. It contains full color page reproductions and full-text search capability, including within ads. Currently, full-text coverage of Variety runs from 1940-2000.
Microfilm coverage of Variety starts with 1905. The call number is MICFILM 651. All microfilm is located on the first floor of the library. If you need to use microfilm, ask for help at the Reference desk; there is also often staff member at the first floor desk who can help you. The bibliographic entries in the American Film Institute Catalog may point you to articles that you can find on microfilm.
Other trade papers are also available on microfilm. Search in the library catalog, discoverE, for our holdings.
Media History Digital Library is an open-access collection of digitized trade and popular publications for film and television. In addition to early issues of Variety, you can find other important publications. Coverage is mostly before 1948 due to copyright restrictions. In particular, look at their Hollywood Studio System Collection (1919-1948).
Los Angeles Times Historical (Coverage 1881-1969). Because of the importance of the film industry to Los Angeles, you can often find articles about film production in the Los Angeles Times.