I've been testing out shooting sports with my 7D. I have a Sony PDX10 that I have normally shot my son's 6th, 7th and 8th grade football games with. It was a solid platform to do that with and its AF system worked really well for that stuff. But since getting the 7D I've only slipped my toe into the waters of sports video shooting. Lots of action to pull focus on seemed to be the hardest thing to deal with at first. So I started to break the events down into manageable bits. Swim meets were the first thing I tried. Focusing only on my daughters lane became easier right away when I used my new follow focus system. I mostly used my 70-200 f4L for swimming. I'm still working on that video.
Then a couple of weeks ago when I was shooting stills at my son's 8th grade basketball game, I turned the camera back to horizontal shooting and shot some of the last few minutes of his game. When I got home and down loaded the footage I was pleasantly surprised by what I saw. The lighting in that gym was horrid! I had to shoot stills at ISO 5000 to even begin to get decent exposures. But shooting video at a SS of 1/50th I could lower the ISO quite a bit. I was pretty happy with my 24-70 f2.8L for shooting the limited part of that game. But I did have to shoot @ f2.8 to maintain a decent exposure. I probably could have moved up into the bleachers and shot from further back using my 85 f1.8, but there was only a couple of minutes left and I had been shooting stills at the baseline for whole game.
So it is possible to shoot sports. Starting with 6 year old flag football will give you a great start. Football is easy because there are these natural start and stop points in the game between action. Soccer and basketball, not so much. The record time of these DSLRs is about 12 minutes of footage before a new file has to be created. So it is possible to shoot until a timeout or and out of bounds play to stop the recording and start a new file.
And as he/she gets bigger and faster so will you with your shooting of the DSLR. I've only been shooting videos with my 7D for about 2 months now and already I am a lot better at it. A knowledge of the game you'll be shooting is very, very important to shooting sports, both for stills and video.
You'll just need to make an investment in at least two 32gb cards, maybe 3. When I shoot my 7D with my new Transcend 32gig 400X CF card at 30fps, 1080p, I only get about 25 minutes of record time on that card. But if you were to get a camcorder that uses tapes, then just think of all the tapes that you'll have to buy over the life of that camera. I can tell you this, I have piles of old mini-DVC tapes. I could have bought quite a few high capacity CF cards with all that money spent on tape stock.