I have four sons and all of them started shooting at around the age of 5. I tried all sorts of stuff, from semi autos (handguns and rifles) to single shots and I've learned a couple of things.
The first things is that it's not good to give a novice shooter a gun that goes bang everytime they pull the trigger without further manipulation. The first reason is that it promotes trigger happiness. The second reason is that in a situation wherein there may have been a mishap of some sort, having a child with a firearm in their hand that is again ready to fire after the mishap could turn a small "oops" into a very dangerous situation.
For a rifle, I went with the single shot chipmunk types of rifles. A few of them have at least decent triggers that won't hinder the little shooter too much. If your dealer has more than one of any particular model, try them all out. The quality of the trigger varies from rifle to rifle even if they are of the same make and model. I followed up the single shot with bolt actions with detachable magazines. The tube magazies are fine, but reloading them is awkward for little ones and can make others at the range a bit nervous.
As for a handgun, try to stick with something that has a decent sight radius. Giving a child a handgun with a sight radius short enough to give a veteran shooter a hard time will only result in frustration and loss of confidence. I went with the Ruger Bearcat.