Nevermind. I just figured out how to get a pet.

@Gingerskye,

Hi there, taking a look at your account it appears that you haven't even created your starter pet yet. To do this please follow along the tutorial which are the boxes on the bottom right of your screen. Dr.Felix and Jinxie will assist you on what to do.

After you've finished the tutorial and get out of Esior's Fields you'll be able to have access to Esior's Haven and you can buy traps there at the market. I would personally wait though, as you'll eventually get a quest from Jinxie with which you'll catch your first pet and she will give you three free traps.

If you do not wish to wait though, you can buy them from the market after the tutorial. To catch a pet, simply explore and find one that you like...then select your trap and click on the pet that you wish to capture and end your turn. The battle log will tell you if the pet was caught, ran away or the trap was unsuccessful.