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:
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.
You can also use a Gift of the Sanctuary to upgrade a pet's rank. This is a special item that promotes any pet you own to the next rank. For example, if your pet is at Violet rank, the Gift of the Sanctuary will upgrade them to Violet+1.
You can use the Gift to grant a rank to any pet of your choice. However, it is more effective to promote a Violet+3-ranked pet. The higher the rank, the more effective the promotion is.
You can get the Gift of the Sanctuary only as a reward in Pet Season. If you do not use the Gift of the Sanctuary by the time Pet Season ends, you will receive it in your in-game mail.
Note
When you collect a Gift of the Sanctuary as a reward in Pet Season, a pop-up window will appear. There, you can select which pet to promote with the Gift of the Sanctuary. You can also close this window and do not collect the Gift of the Sanctuary. In this case, you will receive the Gift in your in-game mail after the Season ends. Once you collect a Gift of the Sanctuary from the letter, it will appear in your Inventory. To find it, check the bottom of the Consumable tab. You can use the Gift anytime; it cannot expire.
If you have collected all available pets and they are already at the max rank, you can exchange the Gift of the Sanctuary for 100,000 emeralds or just wait for new pets and upgrade them.