I started my 9-year old daughter with a .22 cal semi-auto handgun. A BB/pellet gun or a .22cal rimfire were other obvious options, but I went with a Ruger SR-22 for a few reasons.
BB gun/pellet isn't a bad choice -IF- you have a place at home to set up. I could have prepared a place, perhaps, but it would have been a task and there were potential pitfalls. It was actually easier to set up comfortably on a proper shooting range, but it was a private range where one-on-one instruction with no outside shooting noise or people milling about, which is key.
BB/pellet also has another pitfall for a youngster in that depending on the platform, you may only have a gun that requires some manner of a physical "job" that is extremely difficult to accomplish, if even possible. Cocking a single stroke piston or multi-pumping pneumatic can be very difficult for some youngsters. My (now 10) year old daughter couldn't manipulate such a device. Obviously, a CO2 gun is a viable alternative.
Another short bit is that BB guns (as opposed to pellet guns) are absolutely notorious for wild ricochets. That I shot for years as a kid without shooting glasses still amazes me.
I didn't care to go with a rimfire rifle because I don't typically shoot rifles all too often and I wanted to share MY shooting with my daughter, and I didn't have a decent rifle for the job (though I could have bought one, sure) but I didn't have a place with a bench set up and I wanted to focus more on the fundamentals rather than having a small child trying to exercise those fundamentals WHILE holding a rifle as well. I'll readily admit that a rifle is -FAR- easier to keep watch and direct muzzle direction than any handgun ever will be, but I happen to have a
phenomenal student.
And I am 100% on my toes with any new shooter when it comes to muzzle direction. It's one of the most common errors I see with any new handgun shooter.
In any case, I went with =MANY= very short hands-on sessions at home long before we discharged live rounds on a shooting range. As a 10 year old, she handles her handgun dang near as safely as I do, and I'm robotic about it. Her trigger finger is "straight as an arrow" and she does this automatically and it warms my heart to watch it
every single time she picks up or is handed any handgun. Straight as an arrow alongside the trigger guard, it almost looks as though it's been specifically programmed to go directly toward the target and it's simply not allowed to enter the trigger guard and she does it every.single.time. Her finger looks like a bird dog on full point.
So for me, it was as much what kind of gun we started with, it was much more that we did many (short!) serious sessions at home. Many - for the power of repetition. And short to keep it interesting and fresh and enjoyable. I wanted to avoid the pitfall of her getting bored going over the basics.