Cover has to exist before you drive it away
If you already hold a policy, a newly acquired vehicle usually gets a short automatic grace period, with conditions. If you hold no policy at all, nothing extends to the new car and you are uninsured from the moment you pull away. Arranging cover beforehand takes minutes and removes the question.
Check the title before the money moves
Confirm the seller is the registered owner, that there is no lien recorded and that the title is not branded. A dealership checks these as a matter of course. In a private sale nobody does unless you do, and discovering a lien afterwards is a serious problem.
Registration and insurance are separate obligations
Arizona requires both and they are not the same transaction. The vehicle needs insuring in your name and registering in your name, and leaving a gap in either is a straightforward way to end up with a citation or a suspended registration.