Is becoming a script writer a realistic way to break into the entertainment industry? It can be, if you go about it correctly. Writing scripts is not as easy as it may sound, and getting paid for it is even more difficult, but it can be done.
Before launching a career writing scripts for Hollywood, it certainly helps to have an idea of what you want out of your writing career. If you view writing scripts as a way to get your foot in the door so that you can become an actor or professional performer, you're probably on the wrong track. Writing scripts very rarely ever leads to work in front of the camera. Even if you're hoping for a career as a director or producer, script writing probably isn't your best way in. A job as a production assistant is a much better way to learn the ropes of film or TV production. You can then work your way up the ladder to the position of your dreams.
On the other hand, if you see writing scripts professionally as an end in itself - if that's the career you want for yourself - then the best way to get started is by writing. First of all, before writing any scripts, you should familiarize yourself with the format of the media or genres you wish to work in. Script writing for a half-hour television sitcom is quite different from writing scripts for feature length films. There are many books on the market about script format and story structure which can teach you the correct way to approach the type of script you want to write.
Be aware, though: Writing scripts is not a magic bullet. Most literary agents, producers, and studios do not accept cold submissions. That means that they will not even look at a script that you send them, unless it has been recommended by someone they know. And even if your script gets read and liked by the right people, that is far from a guarantee that anyone will pay you for it. It's good to write a couple of good quality scripts and have them in your back pocket to prove your writing chops, but don't expect that you will get paid for writing scripts on spec in that way.
Rather, what you should hope for is a job re-writing an existing script, or adapting a property like a novel or comic book. If your writing stands out and an agent is willing to represent you, eventually you may get hired to do a writing or revising job for a producer or studio. Then writing Scripts will not just be your dream, it will be your job.