Adopted Pets can be customized during the adoption process, however there is no way to change their color or physical features once they have been adopted. You can change your Pet's name for 5 Sapphires from the Pet customization menu located within the animal editor (which you can access by clicking the animal icon on the bottom left of your main screen). You do have two options to get a Pet with a different color: purchase a brand new Pet or trade with another player who has a different Pet.
Some Pets are received by redeeming a Pet Adoption Token at the Adoption Machine on the second floor of Appondale's Claws n' Paws tree. These Pets' colors & features cannot be chosen, and are instead randomly selected.
Please note that if you recycle your pet, you will need to use Sapphires to purchase the Pet again. Some Pets are only available for a limited time and cannot be purchased again if recycled.
Read more about what you can do with your pets in these articles: