Varzok

If you are picking up this quest chain after starting and dropping it previously, you can check your progress by looking at the Taming Kalimdor achievement. The last tamer highlighted on that list ingame (they read left to right, top to bottom, in order) should have the bread crumb quest to the next tamer.

HOWEVER, if for some reason they don't, you can double check your progress by running this script:
/script print(IsQuestFlaggedCompleted(31812))
for each pet tamer in the series. Zunta, The Pet Tamer is 31812, Dagra is 31813, and so on. (You can see the quest ID in the wowhead web address for that quest.)

The last one to give a "true" reply should have your breadcrumb quest. These quest are account wide as of 8.2.5, so it shouldn't matter which character you're picking this up with. You should be able to see a yellow "Quest Available" exclamation point at that tamer's location on your world map.

Here is my Quicklist Guide to the Horde Side of the Pet Trainer Intro Quests. This is a quick reference to see what type of pets you will need to defeat the trainers easily. On Trainers who have the same type of pet for all 3 pets in their party, I would recommend you bring at least two pets that are the opposite type (i.e. bring three appropriate leveled Mechanical types to fight three trainer Beasts). Don't rely on fighting all three of their pets with just one of yours, despite the fact that they will have stronger attacks and defeat them quicker.

1st Tier Horde Trainer Questline (Starts in Kalimdor)

To defeat these trainers with maximum efficiency, you will need pets of the following type:
Mechanical, Beast, Flying, Magic, Aquatic, Elemental

Zunta-Durotar
Quickteam List-Minimum Level 2 Mechanical, Beast
Pet 1: Spike-Level 2 Beast- Use Mechanical
Pet 2: Mumtar-Level 2 Critter-Use Beast

Dagra the Fierce-Northern Barrens
Quickteam List-Minimum Level 3 Mechanical, Beast
Pet 1: Springtail - Level 3 Critter- Use Beast
Pet 2: Ripper-Level 3 Beast-Use Mechanical
Pet 3: Longneck-Level 3 Beast-Use Mechanical

Analynn-Ashenvale
Quickteam List-Minimum Level 5 Flying, Beast, Magic
Pet 1: Mister Pinch-Level 5 Aquatic- Use Flying
Pet 2: Oozer-Level 5 Critter-Use Beast
Pet 3: Flutterby-Level 5 Flying-Use Magic

Zonya the Sadist - Stonetalon Mountains
Quickteam List-Minimum Level 7 Mechanical, Beast
Pet 1: Acidous-Level 7 Beast-Use Mechanical
Pet 2: Constrictor - Level 7 Beast-Use Mechanical
Pet 3: Odoron - Level 7 Critter-Use Beast

Merda Stronghoof-Desolace
Quickteam List-Minimum Level 9 Aquatic, Beast, Flying
Pet 1: Ambershell-Level 9 Elemental-Use Aquatic
Pet 2: Rockhide-Level 9 Critter-Use Beast
Pet 3: Bounder - Level 9 Aquatic-Use Flying

Cassandra Kaboom - Southern Barrens
Quickteam List-Minimum Level 11 Elemental
Pet 1: Whirls-Level 11 Mechanical-Use Elemental
Pet 2: Cluckatron-Level 11 Mechanical-Use Elemental
Pet 3: Gizmo-Level 11 Mechanical-Use Elemental

*Author's Note: In writing this guide, I used various references from wowhead and warcraftpets.com. I did not write this guide as I was doing each trainer. Some information might be wrong. Please feel free to comment and I will change it as needed.

I defeated all of the trainers in the Eastern Kindgdoms for the Alliance side and can't accept any of these quests on my Horde char. I wonder how to get the Taming Kalimdor achiev...