Moving house can be a stressful time for any pet as all their surroundings, sights and smells are suddenly completely changed within a day, so keeping your pet stress free is at the utmost importance, but there are small things you can do to help them, such as on the journey with your pet, try not to put them in an overcrowded car, give them room in it next to a window and some fresh air.
Secondly, put them in the back garden, yard or out of the way, while all the moving off furniture and boxes is taking place, as going back and forth in and out of the house is going to stress them out as they try to figure out what is happening.
Finally, the problem with pets especially dogs and cats are that they are very sensitive to their surroundings and what they think is their territory, so why not take some of the old smells with you? For example if you have a bit of a spare cut-off piece of the old carpet, you could put and hide that in the corner of a room to help the animal adjust, and then after a week or two remove it, the same can be done with old furniture, why not wait a day or two before you give it the big new house clean and scrub, so the animal can regulate the old smell with the new.