If we're talking about a situation where Dad will be taking him out to teach him target shooting, and where the child will never actually be holding the gun without Dad's hands also on it, then I would say that the privilege of learning to shoot a BB gun would be an appropriate kindergarten graduation gift, but it should be clear that the privilege is the gift, and that the gun itself still belongs to Dad.
If we are talking about teaching him to strip and clean it, and allowing him to use it for squirrel hunting and the like, then I think 9-10 is about the right age, presuming he gets formal firearm safety training beforehand. I've personally never heard of a firearm safety class that didn't expect a child to be old enough to know how to read.
I grew up in rural Louisiana, and the rite of passage 40 years ago for boys whose families hunted was getting your own .22 on your 12th birthday. Based on what my nephews got, that still seems accurate. The BB gun of your own usually came 2-3 years before -- but NOT when you were still a preschooler.
At his current age I'd say that a Nerf gun would be the proper speed. He can use that to learn basic safety such as never pointing at a person or a pet, and also not in the direction of any glass window.