The only bug in the experience you describe is that the GM did not offer a Daily quest.
Once unlocked you can travel to each of the Trainers in a region to battle them once each day. The GM will award a bag of pet supplies when defeated, but the other trainers will only provide gold. There is no breadcrumb quest; you have to travel to each trainer's location to accept and complete the Daily.
Personally, I hope they change these up. Maybe after you defeat them, these trainers should travel to a capital city once each day to challenge the player. Or even bring in new NPCs for Daily quest challenges within your capital city: I understand Prince Anduin has become a fairly strong practitioner of the art of Pet Battles since arriving in Pandaria.