Write a "spec script" to send out to publishing and filmmaking companies. A spec script is a completed work --- tailored to a specific screenwriting genre --- that you have available to send after correspondence with a publisher or filmmaker. The spec script should spotlight your writing ability. You should have one for different genres or formats, such as feature-length movie, sitcom, episodic one-hour movie, etc. You should consider your spec script as your personal business card.
Consult a writers' market guide or website to locate screenwriting publishers and buyers. A great reference tool to use is the Writer's Market, available from Writer's Digest books. Freelancewriting.com and Writingcareer.com are also excellent resources.
Create a list of publishers and screenplay producers that you want to send your work to. Pay attention to the guidelines for initial contact. Many publishers respond only to query letters first, and some do not accept screenplays from writers who do not have agents, but many do accept unsolicited manuscripts. Follow the specific guidelines for each company.
Follow up on your submission according to the publisher's specific guidelines. Many publishers and filmmakers give a timetable for responding to submissions. You can send a courteous email or postcard reminding the publisher of your submission if you have not heard anything within that time frame.