I assume you’re in college (not high school—if I’m wrong, disregard this advice)
First of all, have you heard of zipcars? They’re cars you can rent that are available for students at some schools. I think you pay a monthly fee to be a member, and they charge you per hour when you use them. Might be expensive in the long run, but it might also help you get away in the short term while you save up for your own.
Second thing—the nice thing about college is that you can put it on hold. If your first priority right now is getting away from your family, it might be better to be able to work a little more and gain financial independence. True, college can help you in the long run and can give you something positive to focus on, but my impression is that you want out NOW. It’s totally valid to drop it for a while—it’s not “quitting,” it’s about getting your life where you want it. Even if you only stop for a semester. Additionally, it can be very hard on you to keep up with both work and school, and you’re clearly going through it right now. It might be good for you to take a little break from it while you get away from your family.
However, if the thought of dropping it makes you feel super upset, stay. You’re clearly in a rough place right now, and it’s important that you have positive things in your life. If you love school, stay in it. It’s completely possible to do both work and school.
Either way, I think you can make this work. I know it’s a super tough time for you with getting kicked out of your house on top of your mother causing you emotional turmoil. But you sound super capable, despite everything going on right now. Don’t be afraid to reach out for help, wherever you go.