Are you talking for deer hunting? Or just plinking away. If it's just a youngster you can't beat a .22 to start on. Learning markmenship and gun safety would be the first priority. If you're talking deer hunting maybe a .243 or 30-30 to start with. Though I am not a fan of a .243 for deer hunting it is a great caliber to learn on and a perfect example of, " you'd better hit em where it counts." The 30-30 may be a little more forgiving on less than perfect shots. For a handgun I'd have to stick with the .22 as well. I love my little Browning Buckmark .22 and it's really accurate to boot. By far though, whatever gun they choose the first priority is gun safety and knowning every aspect of that particular gun and exactly how to use it properly. When my nephew was 8 I started to take him in the woods with a toy gun. We practiced gun safety every step of the way. Anytime that little wooden gun would be moved in an unsafe mannor I'd take it away and we'd walk straight out of the woods. He eventually moved up to a red rider, unloaded, to a air gun loaded to a .22. Now he's 12 and people can't believe how discipline he is for such a young hunter.