Pets have different ranks: from White to Violet +3. To upgrade a pet's rank, you need to gather special items, pet Stones. You can get them in Adventures or buy them for Chaos Particles. The latter option becomes available after you complete a specific Adventure. The Adventure that you need to beat depends on the item you wish to purchase.
To view what items you need for a particular pet:
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Click Pets.
- Select the desired pet.
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Go to the Items tab.
After gathering the required items and putting them on your pet, you need to charge them. You can use either Chaos Particles and gold or emeralds.