How does buying a car in the USA work for a UK citizen wanting to travel across country?

Question by David W: How does buying a car in the USA work for a UK citizen wanting to travel across country?
I would like to buy a car in May on the East Coast of the USA and drive it to the West Coast of the USA.

When I go to the car dealer, or buy a car privately. What papers / insurance / tax etc will I need to buy to legitimately drive it - and allow my friends to drive it - across the USA.

At the end of the trip, I would like to sell the car back to a dealership - I don't really mind

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