To change your Club name, you must first go to the Club information panel. (Tap on the speech bubble, then on your Club Badge.)

You are not allowed to use inappropriate words in your Club Name, and you cannot have the same Club Name as another Club. You can change your Club Name ONCE for free, and after that your Club Name change will cost you 200 Gems. So make sure you pick your name wisely!

From the same panel, you can change the type of Club you are, the level required to join your Club, your Club Badge, and your Club notice. All of these changes are free.