The first film shot in the Los Angeles area was by the Biograph Company (based in New York), "A Daring Hold-Up" in 1906.
The first studio in the Los Angeles area - but not Hollywood - was established by the Selig Polyscope Company in Edendale, with construction beginning in August 1909.
D. W. Griffith was the first to direct a motion picture in Hollywood proper. His 17-minute short film "In Old California" was released in March 1910, by the Biograph Company.
Nestor Motion Picture Company was the first Hollywood-based studio. They started filming production in October 1911.
The first feature film (over an hour) made at a Hollywood-based studio was "The Squaw Man," directed by Cecil B. DeMille and Oscar Apfel in 1914.
Not till 1915, however, did California beat the East Coast in film output.
(Photo courtesy of Wikipedia. Ah, the Black Maria – the epitome of movie romance!)
*Attributed to me, just now.