How to Get the Design of the Pet Center
To get the design of the pet center you need to talk with an NPC. She located at the NW part of the starter area, just look for a hut, not much far from the gate where you are asked to pay 300 crowns, bit south from The Frantic Torchlighter puzzle map icon.
Once you built a Pet Center you can purchase foods from the NPC located next to the Pet Center, inside your village/city. Depending on the placement on your building she might be hiding in the grass, but she is there. She will also give a quest that teaches you the basics of pet taming. Using foods on animals is useful as first or second action as it increases both Trust and Interest.
IMPORTANT: It is not explained very well during that quest, as the game goes on even if you do not manage to do what it is asked, but….. If you use an ability then you supposed the click once the stars fill up. Hit the mouse button when the second star just lighted up to get a Good or Perfect bonus, that way your used ability count as 1,5X or 2X stronger.
Pet Limits After building a Pet Center you can have 4 pets. You can have only one Pet Center. HOWEVER, you can unlock a building called Pet Shelter. Each of these building increase Pet Limits with 10. Why you need more than 4 storage? Each character can be followed by one pet at a time, but some pets add bonus to this, so eventually, each of your characters can be followed by 10 or more pets at the same time. You can release pets at the pet window, that way you can make a spot for new ones.
There are various pet categories. The First time you tame a creature from that category you will get an achievement. Pets have bonuses to abilities or to your city If it is followed by your character. The strength of these skills is random, so the values not 100% accurate, it is just an estimated value by me after catching 3-5 from the same creature.