I would recommend a 20 ga.
I taught my kids to shot with a BB gun. Started out with safety, then progressed to sight picture, hold aim fire. Once the basics are there, the rest is only a small step from rifle to shotgun.
The biggest factor in introducing kids to the outdoors is success. You want a kid to hate fishing, take him/her brook fishing in the spring. Between the bugs and endless miles of no fish, they will hate all fishing. Fish farm is a way better bet.
Same as shooting. Start with positive results. BB gun, clay targets at 3 meters. Then further, progress to a .22 close range.
When you bring in the shotgun, work on busting a milk jug, watermelon etc.
Then when the wing shooting starts, same thing. All going away, slow and easy.
20 guage will throw a good pattern with light loads and small shot.
If this makes no sense. Grab a .410, a few buddies, and try to bust some clays. You'll be humbled fast, and get discouraged quick. Try the same clays with a 20 or 12. Better results for sure.
That is what will hook a kid quick, bust a few, call it quits and end one a high note.