Pack over a period of time, and try to maintain your pet's normal routine.
Buy a high-quality, sturdy pet carrier. If you have a dog or cat whom you want to keep safely confined on moving day, get a carrier ahead of time and gradually accustom your pet to spending time in it.
Purchase a new ID tag for your pet. As soon as you know your new address, get a pet ID tag that includes your new address and telephone number(s). (Or obtain some other visible form of pet identification such as a collar with ID information imprinted on it or an identification band that attaches to the collar but does not dangle like a traditional tag.) An up-to-date ID tag is a lost pet's ticket home. Keep your pet secure. Also have on hand a recent photo of your pet, for use if your pet becomes lost.
On moving day is very hectic--crates are dropped, doors and windows are left open and strangers come and go with furnishings--which is enough to make any cat's hair stand on end. Place your pet (whether in the carrier or not) in a safe, quiet place, such as the bathroom, so that he or she cannot escape. Another possibility is to see if a friend can keep your pet on moving day.
Pack your pet's favorite possessions, such as feeding bowls, bed and toys, in a separate moving box. You can take these things out first in the new home. Something familiar in a strange environment will help making your pet feel more comfortable, just like a child with his teddy bear. Sometimes one of your old, dirty sweatshirts will suffice. Cats measure territory by odor, so old things that carry your pet's scent help to create a sense of belonging. Be sure to have your pet's regular food and litter brand readily accessible. Moving day would not be the best time to explore a new neighborhood for your cat's favorite flavor of canned food.
If your cat does go outdoors, keep him in for a good while after you have moved as he will surely try to find his way back to the old house. If he disappears, look for him at your old home first. Alert the new owner that should a cat appear at her door, it is most likely yours.
If your new space is very large, start your pet off in just a small part of the house with all of his necessities close at hand. Put your cat and his food and litter box in a bedroom for several days. If all seems to be going well, allow your cat the run of the place and make sure to show him where the litter box is situated, perhaps adding it to the one already in use in the bedroom. Once you feel secure enough that the second box is being used, remove the first one.