First you need to access the Altar of Illusion. Go to the Wizard in his tower, which is located southwest of your farm. You can find it on the left side of the map (marked with a large, blue tower).
To find the Wizard, check the lower floors of the tower, but to talk with him you need to have a friendship of at least 4 hearts with him. To do this, you need to give him the following items:
He also likes Quartz, but doesn't provide as much friendship as the other four options. I will add that the Wizard's birthday is December 17th. Any gifts you offer that day will have an increased effect.
After you have a sufficient level of friendship, go down to the basement and use the Altar of Illusion. Stand in front of him and offer 500 gold coins to change your appearance. You can change several basic characteristics of your character + you have the ability to change the types of animals roaming your farm.
You should have several options for your cat and dog.
