Age, when the child is self disciplined enough to obey the Four Rules and strong enough to hold up a light shotgun. I met one 8 or 9 year old recently who was shooting with his grandad. He used an 870 Express YE in 20 gauge to good effect. Others will not be ready at that age.
As for equipment, there's lots of choices. One that isn't a choice,IMO,is a 410. While 410s tend to be light, the picayune shot charge leads to more frustration than most kids can handle.
A cheap single shot in 28 gauge might be the best choice. Simple MOA, easy to make safe,light enough for small tyros, little kick, and easy to shorten the stock. While the new shooter may want to move on to a "Bigger" gun afterwards, the little 28 will be there as a loaner, "rain gun" or for the next generation.
Other choices would include the youth model pumps, the excellent 20 gauge 1100, and small gauge doubles like the Savage 311.
Start off with the lightest loads you can get.