This is supposed to be a comprehensive list on how to spec your pet (resistance wise) for all the 10- and 25-man raids you're going to see. Please add missing entries by replying so I can edit them in.
Overall, Arcane and Fire resistance help you the most in many of the earlier instances. So if you can't be bothered retraining your pet(s) a lot, keep at least one pet with those two resistances.
Karazhan
Arcane resistance for Curator and Aran, as well as some trash mobs
Fire resistance for Nightbane (important!), also helps on Aran
Fire and Shadow resistance for Prince
If you don't want to retrain the pet I recommend maximum fire resistance if your guild is doing Nightbane, since your pet will die far too easily without fire res (assuming that you want to maximize the DPS time of your pet).
Max out Arcane resistance as well.
Shadow helps on Prince but is not essential since Avoidance Rank 2 will already mitigate enough of the shadow damage so you can easily heal through it. I recommend having Fire resistance for the Prince fight, since there will be delays until the MT moves out of Infernal AoE so your pet will likely take a fair bit of fire damage no matter what.
Zul'Aman
Nature and Fire resistance
Fire resistance for Jan'alai
Shadow resistance for Malacrass
Your pet will take a fair, but manageable amount of Nature damage from bosses, and some of the trash deals fire damage, so it's handy to have resist against that as well.
Gruul's Lair
Fire resistance can help against the AoE mage, but I recommend speccing max armor (first priority) and stamina (second priority) for the Gruul fight, since all his damage is physical. Gruul's "Cave In" ability is a high threat to your pet, so do whatever you can to make it live longer by speccing max armor.
Tempest Keep
Arcane resistance for Void Reaver (essential)
Arcane resistance for Solarian
Fire resistance for Al'ar (you will still see your pet die a lot, this is a very pet unfriendly fight; still do what you can and max out Fire res, it shows)
Serpentshrine Cavern
Nature resistance
Shadow resistance for Fathom Lord