Watch plays and read play scripts. If you want to become a writer, you read and write. If you want to become a playwright, you watch plays and write plays. Live the life.
Search area colleges for playwright and theater programs. Take any creative writing classes that are available. Take acting lessons as well, and learn what it takes for a playwright's work to translate into a play.
Study dialogue, including general conversations around you and dialogue written by any playwright whose work you enjoy. Dialogue is hard for most fiction writers and is no easy task for playwrights either.
Write as often as possible. Study plays of other playwrights and try your hand at a few yourself. Start with a one-act play to get a feel for the scriptwriting process.
Submit samples to local theater groups to garner interest in your plays. A particular group may welcome a new playwright into the fold.
Read trade magazines and newsletters in the industry and network with local theater groups. It's helpful to know the individuals in your area who are on the pulse of the workings of each theater in the area.
Research theatrical publishers and submit a play script to the right one. Check out publishers with similar plays to yours, then research their websites and follow their submission guidelines closely.