The Pet Wardrobe is located in the Pet Yard and can be used to change the appearance and family of your pet. The Pet Wardrobe menu can be opened via the Console in the pet yard, pictured above.
When you open the Pet Wardrobe interface, you can see the pet that is currently following you on the top left. You can switch this pet by clicking on it.
On the right side you can see the pet skins you currently have unlocked. The ones that are not unlocked will be greyed out.
When you select a skin, it will appear on the bottom left, below your selected pet. From there, you can use either Realm Gold or Fame to change the skin and family of your pet.
Do keep in mind that Pet Family matters when changing your pet skin. If you select a pet skin that is in the same family, the cost of changing will be 20 Realm Gold or 200 Fame. If you select a pet of a different family, the cost will be higher - 100 Realm Gold or 1000 Fame.