There are two directions
- Collecting
- Battling
Collecting
Collectors …collect. We all collect, to one extent or another. You choose your collection goals. Some people collect all colour variants of the species.
About half of all species are wild, that you capture in battle. The other half are from drops, achievements, vendors and so on. Mostly, the drop pets can be caged and sold on the Auction House, which means you can buy them.
For capturing wild pets, having a pet with Superbark, like Grumpy, or Weakening Blow, like Terrible Turnip, is important. I find Howl+Superbark from Grumpy great for quickly taking a target pet’s health down to capture range with no danger of killing it.
The addon PetTracker will be a great help with finding wild pets.
For collecting, you follow the Safari achievements for wild pets, and the Raiding with Leashes to start with drop pets. You get credit on Raiding with Leashes even if you buy those pets on the AH.
Battling
PvE battling is, I would say, 99% of the battling game. Simply follow the battling achievements
Follow the Taming Kalimdor/Eastern Kingdoms chain through Outland, Northrend, Cata, Mists until you get to Taming the World and get the Safari Hat.
There is a Pandaren Spirit Tamers achieve you pick up the quest for in Shrine after that.
Then tackle Taming Draenor and Tiny Terrors of Tanaan.
After that, the battles switch to World Quests.
- Battle on the Broken Isles will take some time, because the battles are not up every day.
- Battle on Zandalar and Kul Tiras
- Battle in the Shadowlands
- Battle on the Dragon Isles
- Battle in Zaralek Cavern
- Battle on Khaz Algar
I’m sure I’ve missed some of the minor ones.
At this stage, if not earlier, you should visit the Pet Shop in Legion Dalaran to start the Pet Dungeons, beginning with Wailing Critters.
If you choose your pets and Breeds well, you should be able to do all that with about 50 pets of Blue Level 25.
OK, now having done the easy circuit, it’s time to start the harder stuff, the Family achievements, starting with the meta-achievement Family Familiar, which requires you to beat all the Legion WQ using only teams made of each Family.
This will require a lot more pets, and more attention to Breeds.
I think that’s enough for now. 
You NEED the addons Rematch and Battle Pet Breed ID.
You should pay attention to the Breed of the pets you collect that you intend to use for battling. This will explain why,and which breeds to choose for your main battlers Pet Breed Survey 2019