Pretty good advise so far.
I would recommend buying a quality .22 bolt action rifle and do a few controlled range visits when they are able to listen. Safety is the main message. Handling safety first them basics of accuracy. I bought a Ruger77/22 for my daughters it was the best firearm purchase yet. Slow controled fire
Ruger® Rotary Magazine 77/22® Bolt-Action Rifle Models
I have seen yard air rifle exposed kids that have very bad handling habits. Their Folks just bought them a crossmanand let them go.
Kids also like reactive targets to keep things interesting. Gophers Work great.
My daughter is proficient with .45ACP- .308 at 13