Since I run a state-licensed cat rescue, I often take part in transporting kitties from vet appointments, other facilities, and have even travel out of state with a rescue kitty. Out of everything I have tried, I am listing the things that have helped the most, and hope that they work well for you too.
1.) Get a kennel, like a kennel for small dogs. I use this to transport our kitties in because it has more room in it than a standard cat carrier. Plus, if the car ride is a long one, your kitties will want to stand, move, and stretch, all of which they can do in a small-dog kennel. A kennel is also easier to buckle into place than a typical cat carrier.
2.)Place a small litter pan in the back of the kennel. This is another good thing about a kennel; it has room for the kitty AND their litter box, unlike a typical cat carrier.
3.) Do not let your kitty have access to food in the kennel, as kitties often become car sick and can vomit it all back up. Instead, when you take a break from driving, offer a small bite or treat while resting. Water is okay to have in the kennel, and they actually make bowls that attach to the inside of the kennel nicely (Walmart has these).
4.) Get a kennel cover. These are made to fit perfectly over the kennel, making your kitty feel more secure. If your kitty can see outside (like in a cat carrier), they get nervous, meow loudly, can become sick, and vomit/bathroom on themselves. Cats naturally like to hide in dark places, so a kennel cover makes it easy. Plus, you can always zip open the front part of the cover to check on your kitty at any time, and your kitty will be able to breathe just fine with the cover on.
5.) Put a blanket, pillow, or toy in the kennel that your kitty is familiar with. The smell of something familiar will also help calm them down.
6.) It is best to introduce your kitty to a home after all the old furniture, etc, has been moved into place. This way, the kitty recognizes all the same smells, which help comfort him. If this isn't the case, then that's okay too. Try and show him/her where the litter box and food/water is first, and show them again 2-3 more times that day so they get used to the new location of their stuff.
Good luck with the move and with your kitties!