A pet can evolve at 2 different grade levels 12 & 18. This takes a long time to do, so do not expect to have a evolved pet in a week of playing. You can check a pets grade level by mousing over the pet in the pet "Stats" panel (press "P" then click the "Stats" button).To increase the grade of the pet, there are three things a player can do:
Use the pets skills (slow method)
Upgrade its skill growth by obtaining "Free Lunch" pets and leveling these to 30, then releasing - you will receive a training jade to use on your main pet
Upgrade its skill growth via celebean trade (fast method)
Upgrading all available stats is the fastest way to get a pet to grade 12 (rather than focusing on one or two, like Health or Evasion). Getting a pet to grade 12 upgrading the same stat is possible, but takes a lot longer to do.
Use the training jades at Pet Master Ellan by removing your pet and using the "Train Pet" option. Each of these will upgrade 1 growth stat. Once each growth stat is at or near 1300, your pet should be grade 12.