Once you've reached level 21, you can join a Club. Once you've reached level 31, you may create a Club.
If you're not a member of a Club, tapping the Club button will reveal the following menu:
You can find a specific Club by searching for its name. Or you can see what the top Clubs are and try to join one. Or the game will provide a random suggestion for a Club you qualify for and can join if you’d like! You may also create a Club yourself.