Absolutely! Many sorts of people came West, including those who were educated, but educated or not, people were always interested in a good show. Tom Maguire opened his Maguire's Opera House in San Francisco in the late 1850s and it was an immediate hit. He brought in nationally-ranked acts. When he opened its sister house in Virginia City, Nevada in 1863, there were prominent actors (including the eldest of the Booth brothers) there to perform. And actors continue to come, playing the "circuit" from theater to theater. They offered a wide selection of plays, from light comedies to serious theater, often drawing on Shakespeare. I deal with this in my book, Ronald M. James, Virginia City: Secrets of Western Past (University of Nebraska, 2012).
As /u/itsallfolklore mentioned, the more settled parts of the American West had theaters that boasted traveling celebrities as well as permanent stock companies. These cities with permanent theatrical spaces and railroad infrastructure eventually gave rise to the Theater Syndicate in the 1890s and world-class entertainment was available from coast to coast.
The other side of the story is the traveling itinerant companies who took theater deeper into America's hinterlands. Out there towns were small, competition was scarce, and entertainment was in high demand. The companies that hopped between these villages were tightly knit communities of performers and were often families.
Often these towns had no permanent theaters, so the actors had to be very versatile. They performed in saloons, meeting houses, private dining rooms, barns, or even the open air. Every actor in the troupe had to also be a stagehand, director, musician, and publicity agent.
The textbook example of such a troupe is the Chapman Family who performed many plays in repertory, including works by Shakespeare. The Chapmans pioneered the concept of the showboat (warning: PDF) in the 1820s and later barnstormed California's Gold Coast.
Another helpful book on the subject is Actors and American Culture, 1880-1920 by Benjamin McArthur (University of Iowa Press, 2000).