This will help prevent car sickness. Bring along a bowl of ice cubes so the dog will have some cold water to drink along the way. Stop every two to three hours to walk your dog. Allow them some time to walk, stretch and relieve themselves. Always, always, always keep them on the leash.
Prevention is the key here.
Never take your dog off the leash. Make sure the dog's tags are current. Make a temporary tag with the local information of where you are staying including dates and phone numbers. The National Dog registry uses an identifying tattoo to identify found pets. Cats should always travel in a cage.