A great place to start your hunt is the powerful in-game search tool. Find it by clicking your XP Level number on your home village screen, then the “Clans” tab, then “Filters”.
There you can sift for suitable Clans by setting what you want from a Clan. This includes basic things, such as their location, and more advanced info, such as how many Clan Points and Minimum Clan Level they should have. Clan Wars are a big part of being in a Clan so be sure to narrow down the War Frequency to what suits your needs and/or you know you'll be able to participate!
And don't forget to set the "Only Clans I Can Join" to green.
While you’re in the game, keep an eye on the Global Chat. Clans often use this feed to announce they are looking for new members. Simply click on the player's message in the chat to "View Clan" and easily check what their Clan is like!
Alternatively, you can also advertise in the Global Chat that you are looking for a Clan.
Clans also recruit new members in the big social channels, so don’t forget about them! They often make their posts as tempting as possible, and they’re always fun to read! And don’t forget Discord, which is a busy recruiting space.
Here's a few links to some key social channels to get you started:
Official Supercell Forums
Clash of Clans Twitter
/r/ClashOfClans Discord Server
When you find a possible Clan, make sure you check their Clan profile. Go to the “Clans” tab and enter their tag in the search box to bring up their details. Here you can see useful information like the levels of the existing Clan members and their Clan description. Some Clans like to keep it very casual, some are ruthlessly ambitious in War or Trophy pushing.
Choose the right type for you, and you’ll progress fast. Good luck, Chief!