My children have been in karate for 2 years now. It's the best money I've ever spent on them. DD, who has always been clumsy and uncoordinated even after 9 years of dance lessons, has blossomed into a graceful confident person. DS finally has a good outlet for his energy!
You've gotten some good advise - epecially Cheerful's advise about 'belt factories.' Any good school will be able to show you a printed test of what is required to pass from belt to belt. They will also allow you a month or two introductory program to try it out before signing a contract. I wouldn't say no just because a place asks for a contract - they have to guarantee they will have income to pay the instructors and keep the lights on. Our studio has a one year contract requirement. After the first year, you can re-sign, quit, or move into a special club and lock in rates for 3 years.
Which brings me to my next point - what else does the studio offer besides basic karate? Can the kids take special weapons classes after the first year? What is the cost for that? (in our studio you have to join a club which is a one time fee paid annually or every three years if you sign a 3 year contract. but you get a cool black uniform!) After the first year, our kids were also eligible to take judo, sparring, and tumbling classes.
How to choose a studio? By far, the best way is to go to a karate tournament. Tournament season is just starting, so get on the internet. My kids/studio only participates in NASKA events. Go to a tournament and spend the day watching different age groups and types of events. You'll see the differences in abilities and behaviours from different studios. A studio that rushes kids through belts is not going to be as competative. Observe how the parents are behaving. Are they acting like jerks or are the supportive and displaying good sportsmanship? Make note of the studios that stand out to you and go talk to them. All of these tournaments are open to the public and usually charge a small admission fee. You'll be amazed by the energy in the rooms and the wide variety of events kids can participate in. Don't be overwhelmed though!
Finally, don't pick a studio just because it is convenient. I drive a long way to our studio 3x a week. I could go to one closer but it would be a waste of my time and money - be better off not doing it.