When you've defeated enough enemies to get the pet to level 2, return to Lily White (265,129). She will send DFO Gold you to Master Zhao (160,157) so that you can receive yet another recommendation notice.
Master Zhao (160,157) will give you a quest that is similar to your first kill quest, except this one sends you to the east; it's called "Kill More." You will need to travel east of Da Li to Mt. Infinity to kill 8 Blue Spiders. Once you've slain 8 spiders, return to Master Zhao (160,157) for your reward and next quest.
"How to Catch a Pet" is a quest that introduces you to capturing animals in the wild in order to claim them as your allies. Find Lily White (265,129) again and she will tell you how to capture a pet for yourself.
After you read her instructions and accept her quest, proceed to Serious Wong (275, 49) who will give you entry into the back garden so that you can capture a pet of your own.
When you are done capturing your duck, return to Lily to complete the quest, if you ever need any supplies for said duck or any of your other pets, you can return to Lily at any time.
Return to Master Zhao (160,157) for your reward and your next quest which involves in game announcements.