I want all pets to have access to the same bonuses..that way we don't have everyone with Baby Behemoths because they have the best bonus. Maybe have a tiered levelling system, with say 3 options per tier.

Tier 1 could be:

Pet automatically holds all Materia for you.
Pet has a quiver, holds extra ammo.
Pet does a trick (Could have 3 useful abilities, but some people would probably like to level up pets just for vanity).

Final Tier could be things like taking items to a vendor for you, or allowing access to retainer in the field.

Bonuses would be pet specific, so you could have a different pet for different occasions..
I'm chilling around town, I want to use my vanity pet with fun abilities (Bomb) VS I'm going to level archer, I want my quiver pet (Behemoth).