Starting with BB gun is best, IMHO. My parents were both avid bird hunters and rabbit hunters. They started me out that way, too. We have NEVER had a firearm accident. Safety and respect of firearms first.
I have helped my son learn marksmanship with his BB gun (he's 10 now) and let him shoot my AK recently. He's still too small to hold it on his own but braced on my truck bed he shoots better than me! He will be 11 before this season, and I bought him a Rossi .223 single shot youth that he can handle.
As for marksman training, BB's are not that accurate. I think a good idea would be to go to a pellet rifle after the BB gun. More accurate and you can mount a scope on it to get him used to using a scope. Also, I started him target practicing rested. Then, I would catch him while he was playing hard and had his heart going real good. Have him shoot some while breathing hard so he could learn to control his shot under stress (like when that big buck is within range). Like I said, he's now a better shot than me. LOL
Good luck to you both!