"Lets say you are going out to buy your first car (so you have no motor insurance). You pick out a car at the dealership and its ready to drive away. How does the insurance work?"

Chances are, if its your first car, you are a very young person, and you should be able to have the car covered on your parents policy with a phone call to their insurer before the car leaves the lot. If the insurer requires the policy won't become binding until midnight the day of the call, you drive it home the next day.