Get your car insurance updated. Registration requires proof of insurance in the same state you are registering the car in, so this is the first step. Call your insurance company, in some cases you may have to cancel your old policy or even find a new insurer. You should be able to accomplish this with one phone call.
Every state requires you to re-register your car once you move (if you are staying in the same state you just need to change your address). You will have about 30 days to re-register your car depending on the state you are moving to. Call the local DMV to figure out which documents you will need to bring (title, proof of insurance, etcetera). You may have to pay a fee but you’ll walk out with a new license plate, check with your old state if you need to send back your old license plate. The key to any DMV trip is preparation, know what you need ahead of time so you don’t have to make another trip.
You will also have to apply for a new license. Every state is different but you will have roughly 30 days to get this taken care of also. If you do not, the state will treat you like an unlicensed driver, a serious crime if you get caught.
Like with most things, and especially with moving, make sure you are prepared. You have to know what to expect.