I make a living off of writing and a good part of that is scripts. I had been writing and marketing my scripts for 20 years before I got to this point so it's not a quick or easy process.
Step 1: Write A LOT! - Don't be afraid to write something bad. I wrote a lot of bad scripts before I wrote good ones.
Step 2: Enter contests - No contest is too small. Don't just go for the big contests that promise fame and forture. Go for the local, small contests. I got my start with small contests and got great feedback from these.
Step 3: Learn how to market your own writing online. I did a lot better promoting my own writing rather than going through traditional publishers and agents. I've done both and I've made much more on my own. This may not be true for everyone though. It is good to try traditional means too. Some writers still make a living the old fashioned way.
Step 4: Learn from other writers and make connections to stage theatres and/or movie makers (depending on the scripts you write). Don't just focus on Hollywood or Broadway. You learn a lot from small community theatres and new movie makers. Start small and learn as much as possible from the people who use your scripts.
Step 5: Have fun! If you don't enjoy your own writing, no one else will enjoy it either. You have to be your own biggest fan. That love of your writing will come through.
Here is a great script writing video to get you started: http://freedrama.net/writing.html