Give the cats a large collapsible kennel (I used the fabric and plastic tubing type). It keeps them out of your driving space, and keeps them safe in one spot. They'll likely spend most of their time lying in the litter box, so keep it clean when you stop at rest stops, and always offer them food at the rest stops. Water should be always available and in a reasonably-sized bowl/cup that won't slosh about easily.
Stop at every rest stop, too. Even if you don't need to "go", you still need the break to stretch and give your eyes/mind a break from driving.
If you need to change interstates or go on state routes, check the actual set up of the exits. This always got me when I'd drive through Memphis and Kansas City because their exits are on both sides of the road (left and right).
When you're tired, go ahead and call it a night and find a hotel. PS - make sure they're Pet Friendly or at least have a reasonable pet deposit.
South Dakota is made to induce Highway Hypnosis. Take a break every half-hour if you need to. And stop by Wall Drug - the kitschiest tourist trap in America, but it has 5-cent coffee and is worth a lookie.
Get your car thoroughly tuned up and all the fluids replaced - including getting the differential greased (not checking that caused my car to catch fire in Georgia).
I advise to not move in the winter - it becomes too complicated.
Price your options - it may be cheaper to ship your stuff by freight than to rent a U-Haul.
Have a First Night Box ready with toiletries, bedding, towels, lamp, etc. so you don't have to unpack everything as soon as you get in.
Don't be surprised if homesickness hits and you wonder WTF you were thinking by moving. It'll pass.
Take your time and enjoy the trip. I regret not being about to do that when I had to move cross-country twice.