You can choose the skin of your pet completely free of charge when you reach Intimacy level 10. It’s a pure cosmetic change as in terms of stats, they remain the same as when the basic skin is chosen.
In my opinion the fastest and cheapest way is to purchase a few Green Apples through Exchange and gift them to your Poring (click on pet’s icon and choose Gift). Green Apple is 8-12k zeny/each and you will need 6-8 of them to get Poring from intimacy level 1 to level 10.
If you prefer to do it the cost-less way, please check the guide below.
Check: How to increase Pet Intimacy
When your Poring is level 10, click on its icon (it’s in the top-left corner, below the character’s icon), then Check. The pop-up with its stats will show up. Click on Poring avatar in bottom-left corner.
You will be able to choose your pet’s new skin!
That’s all! Enjoy your pet’s new look.