To give your pet a different colour, you will first need to have a Fur Tonic in your backpack.
There are different places to get a Fur Tonic:
When you have the fur tonic of your choice, you go to the Grooming Parlor.
You select the pet you want to groom. On the next page you will see a "machine". This machine is on the current fur tonic.
With the arrows on each side of the picture of the tonic, you can go to the tonic you have in your backpack.
Once you have found the tonic, click on Activate. This will turn your pet into the colour of the tonic.
To find out how a pet will look click here
Once you have activated a Fur Tonic, you can go back to other "unlocked" Fur Tonics. The background of the tonic will be green.
Vampire Tonics work differently!!! Read here about the Vampire Pet.