1. Shop around and chose the cheapest place rather than the safest.
Martial arts and BJJ can be dangerous if not properly taught. Some important things to consider is finding out who will be your instructor, their teaching experience and methods, who designs the curriculum for the kids and for the adults. Is the place clean? Is the mat area safe? Everyone can lead a class but not everyone can make you or your child better.
2. Underestimate the environment
Look around when you are inside a prospect school and check if you or your child would relate to the people that train there. If the language is appropriate for yourself and child, what type of music do they play out loud… (Usually, no music is played in Martial arts). Would men hitting on your wife or daughter always be a concern at that place? Are the coaches role models for your child and yourself?
Are you going to be comfortable with the atmosphere?
3. Underestimate the instructor lineage and also overestimate his lineage.
Lineage in Jiu Jitsu is very important. Knowing who your instructor learned from, who promoted them to Black Belt, and if the lineage is authentic.
Sometimes the lineage is great but the instructor has no good values and morals. So you need really to do your homework and check their backgrounds and find a balance. 
4. Not knowing the difference between a gym and a school.
Usually at gyms you just rent their facility and pay for it with little to no instruction at all. At a school of Martial arts or fitness you will have your classes designed by the head coach with a direction to take and a culture of respect and values to follow. You might have classes at gyms but there is no direction or culture behind it. Often times, gyms hire coaches from different schools and different beliefs and place them in the same mat or classroom. Gyms tend to change coaches often and with that the culture and progression of learning can be shifted negatively. Also very common gyms hire people that are not black belts but claim to be. You can Google it and see the cases of fake Brazilian Jiu Jitsu black belts.
5. Overestimating the fighting aspect.
When looking for a martial arts school you are definitely interested in legit martial arts teachings and a legit school that will teach you or your children really how to defend themselves. This is not only it, because the instructor can be a great fighter or even great teacher and makes your child and/or yourself a champ but he could also not be a great role model. It is sad but unfortunately there are many cases of Martial arts instructors and BJJ instructors being child molesters, thieves etc.
6. Ignore the true Martial arts spirit.
There are many styles of fighting around the world, some that many people have never heard of. But just a few became true game changers in society like judo, wrestling and now Jiu Jitsu. True Martial arts is not about beating up people, feeling tough, or being a meathead. The Martial arts spirit is to be proud and confident for sure but also to help others also be proud. The spirit is to impact our society making it a better place to be in. “Progress and Mutual benefit”.