Pets are a side attraction in Pillars of Eternity. They were designed for the players who wanted to have their own animal friend within the game. The important thing to know is that their only role in the game is to follow the main character and to look cute. They don't have any other functions. The pets, in contrast to rogue's animals or summoned creatures, don't take part in any of the fights. They can't help you in eliminating your enemies, but you also don't have to worry about them being targeted by the monsters you encounter in the game. You can have one pet at a time and you can "activate it" by dragging its icon to the correct slot in the main character's inventory window. From this moment onwards the pet you've chosen will be visible on the screen and will follow you everywhere you go.
The table seen below lists all of the locations where you can find new pets. Some of them are typical companions (a cat or a dog), however you can also adopt more unique creatures like a miniature version of an animat or a tiny wurm.
AREA WHERE YOU CAN FIND IT
Complete the Brave Derrin side quest received from Odda in the Ondra's Gift district in Defiance Bay (M23,4).
Head over to Raedric's Hold. Enter the keep's dungeons and proceed towards one of the northern rooms. It's the room where you can also find Osrya (M10,11).
Go to The Salty Mast brothel located in the Ondra's Gift district in Defiance Bay (M23,5) and meet with Maea.
Explore the Valtas Residence located in the Heritage Hill district in Defiance Bay (M24,4). Do this AFTER you've completed the Safe Haven side quest.
Orange Tabby Cat
You can find this pet near the Builders' Hearth found in the Hearthsong district in Twin Elms (M36,6).
Meet with farmer Rumbald in the Dyrford Village (M29,3) and agree to complete A Farmer's Plight side quest.
Small Black Lab
Find the pet on the upper floor of The Black Hound Inn located in Gilded Vale (M4,2). Once you've said hello to the dog ask the inn's owner if you can adopt it.
Meet with Gordy in the Copperlane district in Defiance Bay (M18,10). You must complete the Something Secret side quest.
Proceed to Cliaban Rilag ruins. You must get to one of the chambers located in the middle part of the first level (M33,10). Kill the monsters in this room and locate the pet.
Meet with Alarhi in the marketplace area located in the Hearthsong district in Twin Elms (M36,5). She will appear in the market AFTER you've completed the Hard Bargain side quest the way she wanted to.
Tiny Obsidian Wurm
This pet is reserved for the people who backed Pillars of Eternity during the Kickstarter campaign. If you supported the game then the wurm will be available from the very beginning.
Go to Caed Nua (M12) and explore the keep in order to locate this pet.
Meet with Iswld in The Golden Grove located in the Elms' Reach district in Twin Elms (M37,5).
Tiny White Wurm
Explore the second level of the Endless Paths of Od Nua and interact with a wurm egg found in the northern part of the map (M42,2).
Find the dead pilgrims in the Woodend Plains (M16).