Too Young
I own a martial arts school and will not take students until they are 8 but may move that back to 10 years old. If he starts at 6, he will drop out before he is 7. If you find one of the thousands of schools that tell you that 6 is a good age, they just want your money. Look around at their instructors and ask how many 6 year old teachers they have. It will be zero.
If you find a school that promotes kids based on tests like running through traffic cones, it is not martial arts, it is simply a daycare. If they give the kids snacks and help them with homework, it is a daycare.
Also, when parents ask me if I teach discipline, I tell them no. I tell them I only reinforce the discipline that is taught at home. I am not the parent, they are. I am there to be a good role model and give the kids a workout but not raise them.
Now, I am going to ask you: do you want your six year old learning how to punch, kicks, and break bones. Martial arts schools that teach real martial arts teach joint locks and breaks. You are teaching them to punch people and kick people. That does not really relate to learning self worth.
Still, I highly recommend it when he is about ten years old and old enough to understand that martial arts are for self-defense and should be technically advanced.
Now, if you insist on enrolling him, find a school with no contracts that is under $50 per month. Check your ymca's, rec centers, and churches for this. Avoid any school that has the $19 for a uniform and a free week. They are doing the bait and switch. Avoid most schools in the phone book. Never sign a contract. Also, if they claim it is "only" $100 per month, check the add on fees. Those are usually over $50 per month at commercial schools.
Avoid schools with "baby black belts." Fifteen years old is the youngest most legitimate schools will promote a person to black belt. If you see an 8 year old black belt in the room, leave. Also, what kind of black belt test could an 8 year old pass? The answer is not a hard one.