Which pet for your class
A pet acts on its own timer during a fight, in addition to your turn rather than instead of it, and rarer pets act more often. Only the pet you have equipped uses its combat skill; the rest of the collection contributes stats. So the equipped pet is a single, deliberate choice about one thing your character cannot do for itself, made once per fight.
That framing is the whole guide, and it is why the strongest pet you own is frequently the wrong one to equip. There are five elements: fire strikes and burns, nature strikes and cuts healing, earth strikes and stuns, water heals and never attacks, light does nothing but reduce damage.
The rule: a pet answers what your class cannot
Adding damage to a build that already wins makes it win sooner, which is worth something but is never worth as much as it feels. Adding the thing your class structurally lacks changes which fights you can take at all. So the question is not which element is best, it is which sentence about your class is currently true: I lose fights that run long, I lose to armored enemies, I lose to things that heal, I lose to two unlucky rounds, I lose to being stun-locked.
Earth is the default answer when you are not sure, and for a reason worth stating plainly: a stun removes an enemy turn, so its value rises the longer and closer a fight is. Every other element is worth a fixed amount; earth is worth more precisely in the fights you were not confident about, which is where a pet slot has anything to prove.
By class
Brawler wants fire, specifically a legendary one, because legendary fire pierces part of the target's resistance when it burns and armored enemies are the Brawler's only real wall. Ranger wants earth or fire: both compress the fight into the window where Winged Feet is still active, and earth extends the window by stealing turns. Warrior wants earth, because Warrior decides fights by small margins and a stolen turn is the cheapest margin available.
Runic Mage wants water or light, and this is the least negotiable pairing in the game: the entire class is a bet on reaching turn ten, so anything that buys rounds is worth more than anything that shortens them. Assassin wants light or water for the opposite reason - evasion is variance, and what kills an Assassin is two bad rounds in a row, which is exactly what mitigation and healing insure against. Mage wants earth or light, since its damage is already guaranteed and its problem is having no evasion at all.
Rarity is not the same as strength
Water is the clearest example and it inverts the usual rule. A legendary water pet heals a share of your maximum health; every lower rarity heals a share of the health you are *missing*. Missing-health healing is weak while you are comfortable and enormous when you are nearly dead, which makes a common water pet a genuine comeback mechanic rather than a worse legendary. In a fight you are losing, it is often the better item.
Elsewhere rarity mostly means acting more often, which matters most for the elements whose effect is a roll rather than a number - earth's stun and fire's burn both land more often on a rarer pet, so the same element feels different at different rarities. Star ranks, earned by hatching duplicates, multiply combat output, which is what stops a second copy of a legendary from being worthless.
The rest of the collection
Unequipped pets still contribute stats, so nothing you hatch is wasted and there is no reason to be selective about what you keep. Every pet earns one skill point per level and a fully maxed tree costs exactly the points a maxed pet has, so max level means max tree with nothing spare - but two of the tiers force a choice between mutually exclusive branches, so two identical maxed pets can still be built differently.
Once expeditions open, breadth starts mattering separately: routes are gated by element and development, so the second-best pet in four elements opens more of the system than one perfected legendary. Build the collection sideways for expeditions and vertically for the one pet you actually fight with, and decide which pet that is before you send anybody away.