BigKid, I'd really like to help you with this. Be prepared, there are a ton of shooting sports, and the competitors of each are always happy to tell you why theirs is the best. Hopefully, the competitors are very enthusiastic about getting youngsters into the sport as well. No matter what you get into, plan to spend some money getting good equipment, but plan to spend a LOT of time wtih your son on the range.
I shoot NRA Highpower Rifle-Service Rifle division, and CMP Excellence In Marksmanship (EIC) matches. The two are basically the same with minor differences. The course is shot with rifles (in my case the M1 Garand, the M14, and the AR-15 are the applicable rifles) at distances of 200, 300, and 600 yards from position, meaning Standing, Sitting (or kneeling), and Prone. There are slow and rapid portions of the course. The NRA and CMP are VERY motivated toward getting "juniors" into the sport. Its been like that for a very long time. There are special opportunities just for juniors and their sponsors (parents most of the time) to encourage Junior participation without costing the families a ton of money. It's still a somewhat expensive sport at the beginning, but luckily once you have the equipment, you shouldn't need more for a long time. Match Directors could easily arrange things so that your son could shoot on one relay and you on the other so that you could share some equipment if that's necessary. Eventually, though, you'll each need your own.
How old is your son? (edit, oh, he's 13! Neat!) The reason I ask is because I don't want you to be concerned that this kind of shooting is "too much" for a real youngster. Take a look at this little girl here...
...her name is Anna Burns and she's 12. She is shooting in Match Rifle class (rifles are practically unregulated as long as they are safe and still fired from position) as she has to shoot a non-standard sized AR (shorty 16") due to her small build. I'm sure that she'll start shooting a full size rifle when she gets bigger.
If you'd like more info, please send me an email at [email protected]
Also visit this website for Houston area match info. You may need to register to get in the forum but its free.