Know how to write a screenplay. This sounds obvious, but numerous would-be movie writers never think it through that far. You need to understand both how to tell a story and how to convey it in cinematic terms. Know how to develop a classic three-act structure, how to present a character arc, and how to reveal key information in visual ways. Understand the structure of screenplays and how they differ from novels or other printed works. And have some knowledge of the process by which they get to the screen: a process which includes shaping and rewrites, consultation with producers, input from dozens of other creative sources and ultimately merging your vision with that of the director and the studio.
Get an agent. Screenplay agents can help you sell your script to studios as well as negotiate for favorable terms on your behalf. Reputable screenplay agents are registered with the Writer's Guild of America (see Step 3), and you should not sign on with any agency that they have not accredited. The agency should also be located in the Los Angeles area if possible, and you should make sure that they accept screenplays before contacting them. Smaller or newer agencies are much more likely to accept scripts from new writers than larger agencies.
Join the Writer's Guild of America. The WGA is the principal union for screenwriters working in Hollywood, with benefits and advantages much like that of any other union. Membership in the WGA is based on "units" which you earn by writing stories and teleplays, as well as working within the guild's jurisdiction for a set length of time. It charges an initiation fee of $2,500, with yearly dues to follow. Associate memberships are available with much less stringent requirements. In exchange, the WGA can negotiate with studios on your behalf, help you network with fellow writers, fight for residuals you may be due for your work, and provide credit, pension and health care services to you.
Work your way up the ladder. Like actors, the notion of a writer being "discovered" and fast-tracked through the Hollywood system is a myth. Instead, look for jobs which help expose you to film production. Studios employ script readers to examine and provide synopses for the scripts they receive, and production assistants and writer's assistants receive similar experience in film production. From there, you can develop and work on your own scripts with the benefit of understanding the process by which they are approved.
Be ready to share credit. Hollywood screenplays often go through dozens of drafts and sometimes other writers are brought in to tweak it. The screenplay as it finally appear may bear little resemblance to the one you started out with. The WGA has strict rules governing who gets properly accredited for screenplays, and you will need to abide by them if you wish to thrive.