I just earned my black belt in Karate last month. My daughter has also been training since she was five and is a Red belt now. We go through National Karate, which has some locations in IL. I also trained at a different Tae Kwon Do school for about five years and also taught children's classes for two years.
Pick several schools in your area and check out their websites to see if they show their schedule of classes and to get a feel for what they offer. Then visit the schools while there is a childrens' class in session. You should feel that the instructor is in charge, yet can allow some fun during the class. There is no ideal student/teacher ratio. I've seen teachers hold total control over a fairly large group of children. That's what you want to see. You need to talk with the head of the school to get details of pricing and expectations.
Different martial arts use different techniques - karate and TaeKwonDo use mostly kicking and punching techniqes, other styles use more grappling on the ground types of techniques. Some, like Aikido, teach how to use your opponents momentum against them. Some stress self-defense more than others.
There is nothing totally good or bad about including sparring and/or tournaments in the program. Sparring is the one part of martial arts that is non-choreographed, spontaneous, and teaches how to really use the techniques you are taught. Sparring has been one of the biggest learning opportunities for me. That's what truly taught me self-confidence. You want to be sure that the approach to sparring, for children especially, is a "game of tag" or "point sparring". The idea is to learn and practice control and techniques, NOT to hurt your opponent.
Any decent school should emphasize to kids that they are NOT to use the skills they are learning for anything other than during class and in an emergency situation - to defend themselves. The schools I've been a part of will kick a student out of class if they find out they've "done karate" on a classmate or sibling.