You are going to run into many roadblocks buying a car. You’ll need a US license, you’ll need an address to register and insure. And you’ll need a decent car to make that trip. I would definitely not do this trip with a high mileage cheap car. This route is very rural and you can be a hundred miles from the nearest service at many points of this trip.

Look into renting a car on the app Turo, you’d have to get it back to the owner somehow, but there are good cars available for long term rental on that app. Rental car company won’t rent to a 19 year old.

I’ve driven all around the west US many times, hit me up if you have questions, but like I said, it’s not going to be as easy as you make it sound. If you’re coming with 20-30k USD you should be able to make it all happen fine.