How do you know which pets are strong and weak against other pets? Well, there are pet family traits and then there are abilities on specific pets; specifics is more advanced and you can look them up per pet here. Family traits are as follows:
Aquatic +Elemental -Magic
Beast +Critter -Flying
Critter +Undead -Humanoid
Dragonkin +Magic -Undead
Elemental +Mechanical -Critter
Flying +Aquatic -Dragonkin
Humanoid +Dragonkin -Beast
Magic +Flying -Mechanical
Mechanical +Beast -Elemental
Undead +Humanoid -Aquatic
What's that mean? Use a moth against a frog but avoid using a creepy crate against that same frog.
Avoid the auction house for the time being. Collect wild battle pets and drop/vendor/achievement pets for now. Use healing pets to help capture wild pets and help level other pets with. Going against the tamers (start quest is in Org/Stormwind) will help level you and your pets, select tamers give bags with useful and useless items in them.
I have my favorite pets, just as all do, find what works for you. My personal favorite is Amethyst Spiderling
what with Brittle Webbing
and Leech Life
, but these are not unique to this pet so you don't have to wait until Pandaland to get a spider with these abilities. Amber Moth
is often mentioned but you can substitute a Fungal Moth
for it, and yes, this is another favorite of mine.
There are several addons from curse that are listed in a sticky on the forums here. I particularly like the Pet Battle Breed ID and Pet Battle Teams.
If you are able to get to Pandaland then the Beasts of Fable surrounding pets are great to level your pets against. Battle Stones have a chance to drop whenever you win a pet battle. The lead (primary) pet type is usually the type of stone you receive (beast and flying are more numerous to need stoning as are more likely to be vendor/drops and unable to be captured as rare).
Why do you want a rare pet? Think of pet rarity as tiers of armor. The rarer your pet the more oomph it has. I've also noticed that rarer quality wild pets fight better/smarter than poor quality counterparts, but that could just be me.
Overall, this is a patience thing. The more patient you are in collecting the less waste you'll generate in collecting.