You have a Level 25 Shadow, so I assume you read around enough to learn about the skip to level 25 you can get from a Garrison?
Anyhow, if you have one Level 25 pet, you can learn other Level 25 pets that you buy in cages.
While there are some vendors, the best buyable pets drop in various places and can be caged by players and sold to other players, so if you want to buy pets, the best place to look is the Auction House.
You could go to the AH right now and buy as many pets as you want - BUT BE INFORMED ABOUT WHAT YOU BUY. Two things:
- Pet breeds matter, and I think you probably don’t even know what a breed is yet. Install the addon https://www.curseforge.com/wow/addons/battle_pet_breedid and look at the breeds it displays and look at recommended breeds for that pets. Some pets are pretty useless for battling when you have the wrong breed.
- Pet prices on the Auction House are entirely irrational. The same pet might be listed for 500 gold one day, and 19,000 the next. Know what prices are good and what are too expensive. Look up prices on theunderminejournal.com
You are trying to do the BfA World Quests? You have kind of jumped ahead. Pet battling is best experienced by learning your stuff from the ground up. You can start with BfA World Quests, but I completely understand how someone trying that would get frustrated. You are dropped in with no background and no understanding.
If you just want a cheat-sheet to get you through nearly all BfA World Quests, without having to learn or understand anything, put your hand in your pocket, go to the Auction House, buy and learn
a) 3 Level 25 Blue-quality Ikky
b) 3 Level 25 Blue-quality Stormborne Whelpling
c) 3 Level 25 Blue-quality Iron Starlette with breeds P/P, H/H, P/S
Those pets will smash though nearly every BfA World Quest easily.
Use Ikky like this: Black Claw->Flock->Savage Talon
Use Stormborne Whelpling like: Arcane Storm->Mana Surge(lasts 3 rounds)->Tail Sweep
Use Iron Starlette like: Wind Up->Supercharge->Wind-Up->Wind Up->Powerball->Supercharge->Wind-Up
There will be some variations depending on the conditions, but that’s the basic formula.
Every battle in BfA can be done with those three pets. Most can be done using three of the same, though which to use depends on the battle. Some will need a combination. A couple of battles may not work first time because of RNG, but will work most of the time.
So if you just want an answer, that’s it. It’s not very satisfying, though, and doesn’t develop your roster for more varied and challenging battles like the great Celestial Tournament.
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If you actually want to get into pet battling and collecting, that is a process that will take months of learning, taming, collecting, refining, and doing achievements.
You should start by continuing your Taming achievements. I see you started Taming Kalimdor but stopped after the first two, Zunta and Dagra, and never went on to Analynn.
You do Taming Kalimdor, then Taming Ourland, Taming Northrend, Taming Cata, Taming Pandaria in order. By the time you have finished Taming Pandaria, if you have done it naturally, you will have built up understanding and a roster that will help you face anything. Sadly, most people coming up now seem to have learned only how to look up pre-made answers on Xu-Fu’s or Wowhead. 
As you do those Taming achievements, you should also be working on your Safari achievements - capturing pets in every continent.
Of all communities in this game, pet battlers are the friendliest and most helpful, so if you want to know, just ask! I’m sure you started off frustrated here because you couldn’t post, but don’t let that spoil your approach.