My family and I moved from Ft. Hood to Ft. Stewart. We are a family of five with 2 dogs. We had one of our dogs in his kennel while I drove. We also took alot of stops on the way and we stayed in Air Force Lodging along the way. It was cheaper and the rooms were pet friendly. It was the hardest move we've ever done since we decided to move ourselves. Plan as much as you can ahead of time.
Congratulation on your first move! My first step is always to talk to your transportation office. They will help you establish a moving timeline so you can determine, when you will move, when you schedule the packers and movers, and can find somewhere to live in your new location. Once you establish a timeline you can start to plan out all of the details.
http://www.move.mil/home.htm is a very helpful site where you can manage your move online and watch some helpful informational videos to hopefully make your move smoother! Thanks so much for posting and asking a question, please let me know how your move is going and if you have any other questions! Good Luck!
If you fly, make sure the kennels for the cat and dog are up to the airlines standards. We ran into a few issues with that moving to New Mexico using United Airlines. Every airline has different standards for how the kennel should be.