I haven't actually been able to do any pet battles myself, thanks to the bugginess and subsequent battle removal, but from reading other people's accounts, reading up in the Journal, and watching streams, I think I've figured something out about the order in which pet attacks are learned. At lower levels, pets seem to gain only the abilities from the left side of the table shown in the journal, without access to the ones on the right. Some seem to think it's a bug, but I don't think that's the case.
I haven't seen any proof of this, but I get the feeling that pets gain the left-hand abilities first while levelling up, then gain the ability to choose between those and the right-hand abilities later. One thing that supports this is that all the pets seem to have for their first left-hand ability "Do 20 (type) damage to target", and a LOT of pets seem to have for their first right-hand ability "Do 20 (same type as before) damage to target, double under certain conditions". No one in their right mind would choose 20 Beast Damage when they could get 20 Beast Damage Plus, so I'm thinking the choice isn't actually available for a while.
I don't know if anyone else has brought this up already, or if it's just obvious and it's only because I'm half asleep that it seems like an epiphany. I'm going to try going back to bed. Insomnia ahoy!