Yes, we have been karting at All Sports for years. I even wrote a blog article about it some years ago:
http://www.dcurbanmom.com/weblog/karting
I cannot recommend the program strongly enough. The kids have to take the two day course in order to get their "license". Then, they can participate in the league or even the open races. My son and I have had a lot of fun racing against each other.
SmielyFace is correct about normal karting which requires you to have your own kart. But, at All Sports, you just show up. They have helmets, suits, karts and everything else you need. It is still fairly expensive, but is a great hobby.
As the next step, I'd also recommend Camp Motorsports:
http://www.campmotorsport.com
This is a summer camp for karting that is very good. They also have a weekend parent/child program.