Welcome aboard! Sounds like an interesting adventure you've been living. We have a member (Mudderchuck) who used to live in California and has now lived in Germany for quite some time. He could probably give you a lot of comparison information for European countries versus America in both hunting and living.
In general, each state sets its' own hunting laws and regulations. Some federal laws can apply on federal lands. It would help if you could narrow down what state you hope to move to someday. All states have a required hunter safety program (to be eligible to get a hunting license) where you can learn a lot of good information and make some good contacts.
It would also be helpful to know what kind of game you're considering hunting, such as big game like deer, bear hogs, small game like rabbits, squirrels, varmints like Rock Chucks, etc., predators like fox, coyote, lions or different birds like ducks, geese, pheasant, quail, etc. You can hunt them all but it helps to get more precise answers if you're only interested in hunting birds for instance and say just birds. Hopefully this is helpful.