Copyright your scripts at the US Copyright Office:
http://www.copyright.gov
The best way to improve your writing is to watch movies, read screenplays, and write them. It's going to take a couple scripts under your belt before you really get the hang of it. Don't fall too in love with any particular script. Some newbie writers don't like to hear this. Yes, you might have a great idea, but lack the skill to execute it properly. That skill only comes with experience. Do the best you can and move on. Don't waste time trying to perfect what you believe is your Golden Ticket to success. It doesn't exist.
If you ask 50 people how they broke into Hollywood, you'll get 50 different answers. Try anything and everything (within reason):
You might get lucky with a screenwriting contest.
You might know someone who has a cousin/friend that works for a well-known actor, actress, or producer.
Research successful movies, in the last 5 years or so, on imdb.com that are in the same vein as your script. Contact every producer you can find on that movie and pitch your concept.
Contact directors you feel would be a good fit for your script.
Convince a manager or agent to read your work.
Even if you're lucky enough to get somewhere with these approaches, they'll often lead to dead ends. It's a difficult business to break into. You'll find yourself sending out hundreds of emails a month and maybe getting a handful of responses, at best. Half will be from their legal departments telling you they don't accept unsolicited material.
Keep in mind, there isn't a huge demand for original scripts these days. Sure, sales happen here and there but most don't turn writers into overnight millionaires. There is a market for low budget scripts though. And streaming has increased the demand for content.
I say this all to say, a lot of things need to happen that are out of your control. Focus on the things you can control: being objective about your work, writing as much as possible, making the next script better than the last. Once an opportunity presents itself, you'll be in a much better position to take advantage of it. The last thing you want is a bunch of regrets because you ruined a connect with a lackluster script or only having one script to showcase your writing.
So get those 3-4 scripts written then start strategizing.
FULL DISCLAIMER: I'm not typing this from my yacht, Cash Money Screenwriter, while two bad foreigns give me a back massage. We might have varying levels of experience but at the end of the day, most of us are in the wannabe writer dinghy.
There are a few cats in the business who post and lurk in this thread. I do know a bit about missed opportunities so I thought I'd share some stuff I learned the hard way.
But hey, if you're that confident about your ideas and ability, go shoot the thing yourself. It's not for everybody, and it certainly isn't easy, but it can be better than waiting around for your career to happen.