I'll tell you what I did... and for the most part it is what I would do again... My kiddos were about that age (6-7) when they started shooting. I told them that I'd start taking them shooting when they could not only recite the basic rules of firearms safety, but could explain to me why each of them was important (I was looking for general understanding not philosophical depth - but wanted to be sure that they understood why they needed to follow them - not just memorize the words). Once we started shooting, they would graduate from one gun to the next based on a couple of factors.
First - was I convinced that they had developed good safety habits?
Second - were they developing competence - or more accurately had they developed enough competence to move to the next thing, and had development slowed?
Third - did they WANT to do the next thing? Were they getting bored with the stage that they were at? Fourth - did I think that they could go to the next level without developing bad habits / reactions (youtube might think it's funny to hand a 65lb 9 year old a 45-70 lever gun with a spicy cartridge in it... but I don't).
The guns we used were the following:
1) I started with a single shot / breech break Rossi 22LR with iron sights (it is a "convertible" that has a 410 shotgun barrel that can be swapped in). I bought this gun explicitly for this purpose - and I used it for my kiddos, some nephews, some adult first time shooters, etc. If you are the guy who takes the family kiddos or other new shooters out for the first time - it is a very nice tool to have. I generally think that this is the very best option for a "1st" gun for a kiddo. You can stand there with a pocket of .22LR and would hand them one cartridge at a time, and when the gun goes Bang, you know that it is inert, so when the inevitable muzzle discipline issue happens after a shot (not saying that you shouldn't be "on top of it" - but if you are working with new shooters it WILL happen at some point) you can both react to it to correct, but also understand that the gun is certainly empty. If you have a bolt action 22lr and you don't plan to be the family "gun trainer guy" - then you might choose to skip this step - but if you are going to cycle through many new shooters in your life - this little gun is worth every penny of the $269 or whatever they go for these days.
2) They graduated from there to a bolt action .22LR rifle with iron sights. Here we started with the same "single cartridge" approach... then a pile of cartridges and them single loading, then loading the tube magazine...
3) They graduated from the bolt action .22LR to a Ruger 10/22 semi-auto 22LR with an inexpensive scope. This did not happen until we'd had a couple of shoots with zero muzzle discipline issues and the good habits were clearly in place.
4) They graduated from the 10/22 to 5.56 AR platform rifles. First center-fire gun and easier on the shoulder than a youth bolt rifle.
5) Then a youth .243 - also single shot breech break "handi rifle".
6) After this the world started opening up a bit. Pistols started with an old .22LR revolver, and then a Buckmark semi-auto 22... then followed the same pattern. They started shooting other "heavier" semi-auto and then bolt action center fire stuff and shotguns as their bodies, desire, relative recoil sensitivity made it make sense. By the time they were early teens they were shooting whatever they wanted to shoot and I felt safer next to them on the range than I did with your run of the mill "adult shooter" next to me...
My 2 cents... YMMV