If he doesn't make a move to leave the room, simply shut the door, and try again tomorrow. When he meets you at the door and sticks his head out for a peek, he's warming up to the idea of becoming familar with his new surroundings. If your cat has been allowed outdoors in the past, don't let him outside until you've been at your new residence for several days. When you're ready, place him on a leash for your first introductory outings. Do this for the first three or four trips outdoors, and then, if he seems comfortable, remove the leash and let him explore the outdoors on his own.
Dogs tend to adapt to moving more easily than cats.
If you travel by plane, a small dog may be able to fit under your seat in his pet carrier. Consult with your vet before the move in order to take the necessary measures to reduce stress. And make sure your flight is a direct one so that your dog's carrier won't be exposed to the elements while he waits for a plane transfer. Call ahead to find a hotel that accepts pets so as to avoid leaving your pet in the car overnight. Birds present a unique scenario for any mover.