I'm a 30 year old guy with a 9 year old that rides. Here's what we are doing (for now) and these are somewhat approximate and from memory (I'm not home right now, and maybe remembering frame sizes, etc. wrong):
I bought him a Haro Revo from a lady in town for $15 (seriously, I was sure it would cost me at least a bit more) and we fixed it up a little with some new parts and adjustments. 20" wheel but a smaller frame (maybe 19"?). He was 7, but now he is 9 an still riding it. He rides it around town and at the nearest track on weekends and stuff. For his birthday this year, I picked up an old Haro (around 2004ish F-series) and stripped it bare. Painted and rebuilt wit spare parts from the house. It's a bigger frame (20.5 I think) and a little lighter, but it wasn't in terrible shape when we found it either. I told him maybe next year we'd pick up something. He's about 5'0" and 95-100 pounds (think built like an average football player).
I'm 30, 5'10" and around 200lbs. I got way out of shape in college and started exercising and working out again when I was about 25. Lost 50 pounds and ran a lot. Rode MTBs and road bikes a lot - raced a few triathlons (sprint and attempted olympic) and a few MTB races (a few XC races, one Enduro festival race last year). Rediscovered BMX/Freestyle a while back. I'm riding an old Specialized Fatboy Vegas for now (bought used off CL on a whim) and it is a 20" wheel with a 20.5 frame I believe. I'm thinking my next purchase will be a MTB, but after that I'll probably replace my BMX (thinking this winter). It's not a bad sized bike, it is just heavy and I'd like something newer/lighter/etc. for next spring to race on. Also, I'd like a longer TT because it does feel kind of cramped sometimes. I also have a GT Compe that I ride at the skatepark, and it's a lot of fun but just not something my son's into yet.
Although I've done it, I'd hesitate to buy something used. So many bikes are abused, but if you find a deal...
Also, getting something to work on together is always a good idea -- if you don't already know, you can learn, if you do know, you can teach him...
Also, in my looking for a new bike, reading the Old Guy's Guide to Re-entering BMX was a good first step before I got too far.