A long ago I've made a list of "classic" double counters: pets that are both resistant and strong against a certain family.
As you can see, I also tried to include opponents that tend to use damage source not related to their type ("jugglers" as a placeholder name), but that gives you, in fact, a hundred of combinations in total.
Still, I didn't hestitate to create a chart of these dependencies. Was seeing it as a nerdy catalog myself, but because of Family Familiar and other exotic stuff, I think I can show it to you.
1. Choose your enemy's pet type from the top row and their main attack from the left column.
2. Look on the place where the type column and attack row are crossed.
3. The tiny silver or graphite symbol in the corner means your counter's pet attack. The coloured symbol means your counter's pet family.
(example: "I need to fight a flyer with a beast attack. This shows me that I need a magic attack on a flyer.")
Note that this table doesn't show you specific pets or tips about countering the ability effects (like stuns).
But who knows? I know that someone is planning more, but I'm not sure I can tell you who, because I don't want to flood them with the messages from the impatient ones.