Only you can decide when you think a child is mentally and physically mature enough to be introduced to shooting. I prefer that they be at least 7 or 8 in most cases for "real" firearms. Low-powerd airguns perhaps a bit sooner, and the "Eddie Eagle" type of safety precautions as soon as they're able to toddle about and respond to language.
I don't have any particular preference when it comes to brand, but over 40-odd years and dozens of newbies, both children and adults, I've formed some definite opinions. All I can say is that they've worked out very well IME.
First of all, I highly recommend the use of a manual action over a semiauto with both children and adults. I also favor something with either a visible hammer or cocking piece even though the action should ALWAYS be open and clear until they're ready to fire at the line.
There are several purely pragmatic reasons for this, IMO:
It adds another means for positive, instant visual verification of the firing status of the weapon relative to conditions at all times.
They help to reinforce safety, handling, operational and firing techinque lessons by requiring that a specific set of steps be repeated in the same order for every round fired. This increases the coach's opportunities to identify errors and correct them before they can become habitual and positively reinforce good practices with praise and encouragement.
"Ball and dummie" drills are an important teaching tool. Manual actions help make these easier to do and more effective for all concerned, IMO.
Single shot or repeater, size is important. Learning new things is hard enough without adding unnecessary complications, like trying to cope with something that's about as tough and awkward to get into proper contact with as a deadbeat borrower into the mix. A bit too small almost always works better than too big for both kids and adults. IME, how well it fits the student is much more important than how it operates when it comes to establishing the basics.
Try reactive targets first, especially with kids. It's been my experience that instant positive feedback builds confidence and helps establish and maintain interest at the critical introduction stage better than paper targets. Kids generally respond quickly and with enthusiasm to success, and that almost always translates into wanting to learn how to do it "better".
I've had good results with several types of longarms. I really like the Chipmunk, Cricket, Savage Cub etc. for very small kids. With larger kids my Rossi M62 pumps (one equipped with a shorter "youth" buttstock) have been big favorites. Along with them, my Marlin 39M and a new little Ruger 10/22 Compact have also been very useful to maintain high interest levels and add new skill sets as they outgrow the tiny bolt guns and become experienced enough to "graduate" to repeaters.
For handgun intros, I like revolvers and prefer single actions for many of the same reasons that I prefer manual action rifles.
Depending on the size of the student I've used a little Ruger NM Bearcat, the Single Six and a 4" Taurus M94 with much success. Personally, I prefer not to introduce them to semiautos until they've demonstrated a firm grasp of the basics of safe handling, satisfactory technical competence and shown an adequate understanding the importance of "fire discipline" with a revolver.
Just my own observations and opinions, FWIW to you. You're doing the best thing you possibly can for the future of our sport and our nation by passing the torch along to the next generation, and I thank you for that.