I moved from Los Angeles to DC with 3 dogs, 3 cats, and a parrot.
1. If possible, put the animals in a kennel or other place while you/the movers are picking up your furniture. They will freak, so it's better to just have them elsewhere. It's better to not have to worry about who is making a break for the door.
2. Take your time. By the end of a 12-hour driving day, both you and the critters will be frazzled. If possible, take an extra day or so. Take breaks to feed yourself and the critters. When moving from DC to OK, we were in a hurry to hit the first stopping point, so I skipped supper for the dogs. By the time we stopped, my normally calm, easygoing dogs were cranky and upset. There was much barking, and then there was scratching from the cats. Better to stop early and have everyone remain calm.
3. If possible, have one person stay with critters at all times. At the end of that first evening, I stayed at the hotel to unload while Mom went off to get food (it was about midnight). The dogs started barking as soon as I left the room. Well, I couldn't have that, so I stayed with them to keep them quiet (which they were so long as I was there, thank goodness!), and that meant that by the time Mom got back with tacos, we were all wound up and anxious. Dogs ate tacos, I ate tacos, and then we finally unloaded the cars. It ended up being a very, very late night.*
4. Make sure you know where you're going once you get there. We ended up having to stay in a hotel for a week (with critters!, in DC!) until the rental house was ready for us. I had to be at work that week. It wasn't pretty.
*There were other issues there, such as that we pressed to make it to that hotel, since we had a non-refundable reservation from Expedia, despite massive thunderstorms in the mountains, and the room smelled like rotten cabbage.