It varies.
In NH, the seller keeps the plates while in WA the tags and tabs stay with the car and the seller has five days to file a notice of sale with the state. That also means that in NH, it is wise for the seller to make two copies of the Bill of Sale and keep one as they would otherwise be held responsible for anything the buyer does with that car until it is re-registered under a different name; a problem that WA doesn’t have.
WA requires all vehicles sold to have a title that the buyer signs over to the seller; in NH, a title is not required to sell any vehicle over 15 years old. In NH, the annual safety inspection expires immediately, requiring the buyer to get it re-inspected (for $35–40) before they can register the vehicle, while in WA the bi-annual emissions test is still valid until 24 months after the previous test. In both cases, the buyer needs to get a new title though.
In both states, the registration expires immediately upon selling the car as that is tied to a specific person and specific car; the buyer needs to register it in their name. NH requires the seller to have plates for the car, though they do issue temporary tags for a nominal fee that allow you to drive the car for 10 days. Handy if you buy a car that you don’t intend to register for a while, like a “project car” or “fixer upper”. WA allows the buyer to drive the vehicle unregistered for five days; a reasonable period of time to get to the registration office even if it’s a holiday weekend but no longer than that.
Insurance does not transfer either, nor can it be transferred. Like the registration, it is tied to a certain combination of owner and VIN; change owners or cars and the policy is no longer valid and thus expires. Of course, if something happens between the time that you sell the car and the insurance company is notified of that sale, you will have a problem. Another reason for a smart seller to have two copies of the Bill of Sale ;) Then again, in NH, car insurance is not required by the state unless you have an SR-22 while most states require proof of insurance to register a vehicle regardless.
Of course other states have other rules, so it’s best to research how your state does things.