I'm in the military and serving in Missouri. If I claim a Georgia residence as my home of record should I be taxed by Georgia? Or am I taxed by Missouri or no one at all?

Georgia.
Military personnel and their families have special rules for their state tax situation. They will pay taxes to their home state even they they are stationed at another state. That is why, many in military claim Washington, Texas, Tennessee and Florida as their home state because these state have NO state income tax.
The family member of the military person MUST pay taxes on income earned within the state borders of the state where the family is stationed.

Thank you very much.