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Alright, first things first, make sure you have the Potion in your Inventory. Once it's there, click on the picture of it. You should get a list of a options, like 'Use on So-and-so'. Click on whatever pet's name you'd like to use the potion on. And.. ta-da! Instantaneous new color! Potions are irreversible, so take caution.

There are also Elixirs. Say you have an Arid Celinox Elixir on hands, and you want to use it on your pet. What is it going to do? It's going to turn your pet into an Arid (colored) Celinox (species). This is also irreversible.

Uh-oh! Made a mistake with a potion? Unfortunately, you can't undo a potion/elixir or get your potion/elixir back. You can, however, buy another potion or elixir and use it on your pet.

The other way to turn your pet into another color or species, is by zapping it with the.. Zapper *evilcackle* You can either use the Zapper v2, conveniently located in Atebus, or the Zapper, which is in the Fair Grounds. The v2 (Zapper v2) costs 15 Wizard Tokens, and you can use it every 2 hours. You can also choose which pet you want to be zapped from a drop-down list. Whereas the regualar Zapper in the Fair Grounds only requires 5 Wizard Tokens, you need to have your active pet be the one that's being zapped. And you can use this zapper every 4 hours.

You might encounter some strange things with the Zapper, however. You're pet can change gender, species, color.. It can have it's minion zapped, and its TC (Treasure Chest) size increased.

Another very awesome way to change your pet is by using the Sex Berry. This will change your pet's gender from Male to Female, or Female to Male. If you're pet is genderless, I don't think that it will change. (Will edit later once I'm 100% sure with this answer.)

Or maybe you like your pets color, but don't want to have to risk zapping it, or buying more expensive elixirs and potions.. Then the answer to this is to use a Miracle Potion II! This will change your pet's species to ANY species you want, while maintaining it's current color! How incredibly awesome is that? xD

The other option is to change you're pet to ANY color, while maintaining it's current speices. This can be done using a Miracle Potion. ^^

For the other two colors, Hydrus and Galactic, you'll need to do something other than buy a potion or zap your pet. For Hydrus, you need to collect a serious of Uncharted Land Maps. (1-9). Once you do that, take it the Games section, then go to Treasure Maps to turn it in. Wait about 5 or so minutes, then go to the explore map. Click on the island just to the right of the Fair Grounds and you'll be able to see Merana. There, you can choose which pet you'd like to turn Hydrus. Once you click the image of your pet, you'll have to do a quest for her. And then you're pet can be hydrufied ^^

If you want your pet to be blessed Galactic, you'll need to do this on the 6th and/or 7th of every month. Go to Atebus, and dig around in the Coda Caves for some Gems (which you can dig for every 46 minutes), then go to Itallo to have your pet blessed. He'll than ask you for a random 2-5 gems. Once you get those, you're pet will be turned Galactic!

Nuuuuuclear pets are obtained with a Nuclear Potion that /does not/ restock! You have to make it at the Genetech Lab in the Darkside. :3
Here's a link to all the known nuclear item combinations so far :D