Air rifle or air pistol shooting is good practice for the real thing and alot cheaper. A few of the main reasons are you learn the basics, trigger control, good sight picture and the big one follow-through. I've seen it many times, learn to follow through and improve your shooting.
I imagine there is some legal air rifle shooting in your state. I've had some fun popping Rats and Gophers. Sure there are going to be restrictions, city state and county.
IMO the first thing to do is to take the safety class and get a hunting license. IMO be flexible I hunt varmints and small game all the time, as much or more fun than larger game. There is only so much room in the freezer and scratching that urge to hunt can be accomplished in many ways.