Make sure you have a copy of your pet's current vaccination certificates and ask your veterinarian for a health certificate as well. If possible, have your local vet provide a referral to a veterinarian in your new area. If your pet is taking any medication, make sure you have an ample supply until you are settled with a vet in your new town.
Some states have border inspections off all animals, some only do spot checks.
These checks are usually done either by the officials of the Department of Agriculture or the Highway Patrol. Some states will check for an interstate health certificate, others rely on the pet owner to comply with their laws and do not ask to see the paperwork. Order new ID tags for your pets with your new address. Unfortunately, many pets get lost during a move and if they have the address and contact information of your new home, anyone finding them will be able to contact you. Some old or ill animals will suffer adversely from the stress of moving.