To purchase clothing and furniture for your Pet, head either to the Sapphire Shop in Jamaa Township or the Claws 'n Paws adoption tree found in Appondale.
Pet clothes are items that can be worn by your Pets on either their head or their torso. Once your purchase some clothing for your Pet, click your player portrait (at the bottom left corner of your screen) to customize your Animal. At the top left, there is a small icon of your pet that will take you to pet customization. Click the little red collar icon, next to the switch pet button (bottom left), and see what different accessories your Pet can wear to look extra snazzy.
Pet furniture is a category of den item that specific Pets can interact with. If you decorate your den with an item that your Pet likes, it sometimes will run off and play with the item all on its own! You can tell which items your Pet likes by if their icon shows on the item in-store.
Read more about what you can do with your pets in these articles: