You need a computer. There's just no way around this. Besides being an indispensable writing tool, it also gives you the ability to surf the Internet and start reading through websites devoted to the craft and the business of screenwriting. Since rewriting is such a key part of the process of screenwriting, being able to save, edit and re-visit different drafts of your script is essential.
Use screenwriting software. You'll do yourself a huge favor by buying one of the popular screenwriting programs (Movie Magic Screenwriter and Final Draft are the most popular), or downloading one of the free ones (e.g., Celtx). The format for screenplays is highly specific and fairly complex. These programs will guide you through much of the formatting required for the industry standard, so you don't have to do the all the margin settings and specific page formatting yourself. You'll still have to build a good story with a strong structure (typically laid out in three acts: set-up, conflict and build of action, resolution). You will also need to learn how to write action and dialogue, the two main components of all screenplays. If you prefer to create the formatting yourself, there are guides and books that will walk you through the process.
Watch films. An invaluable resource are movies themselves. Familiarize yourself with award-winning and highly acclaimed films. Take a look at the American Film Institute (AFI) list of the 100 Greatest Films. Learn from the best. Take notes about setting, character, plot, dialogue and action.
Read screenplays that have been produced. Go to the library, bookstores or look on-line for professionally produced scripts from films that you have access to. See the film, then read the screenplay. Pay close attention to how the writer conveys in words what you saw and heard on the screen. Make sure you're not reading a shooting script. A shooting script is the final script that details specific camera shots for a director when the film's ready to be shot. You'll know you are reading a shooting script if there are phrases like "Closeup," "track" or "pan."
Read screenwriting books. Books can go a long way toward helping you understand the technical aspects of screenwriting. You can check with your local university to see what film professors currently recommend. Some screenwriting styles change over time. The way a script like "The Maltese Falcon" was written is very different from "The Bourne Supremacy." You want to find out what the most current styles and formats are.
Take a screenwriting class. A screenwriting class is a hands-on, intensive way to learn how to write a script. There are classes taught at universities, at film schools and on-line. Options range from full degree college programs to three-hour screenwriting intensives.