To join a Clan, you have to be at least Star Level 7. Once at that level, you can go to the Clans section of the game by tapping the Clans icon from the main menu screen. You will be given the choice of joining an existing clan, or creating your own clan.
Tap "Join a Clan", and you can browse through some clans that are looking for more members. If you’re looking for one clan in particular (if your friends have already started a clan for example), tap "Search Clans" and you can search by Clan name/ID, location, type, and number of members.
Once you've found a Clan that you're interested in, simply tap the "Join!" button on that Clan! If the Clan is Closed instead of Open, there will be a "Request" button instead. When you tap that, a request to join the Clan will be sent, and need to be approved by the Clan's lead or a co-leader before you can join.