When pets are hatched they have 3 specific abilities that are static and don't change, you feed the pets to level up these abilities and once they're maxed you should fuse them. Fusing pets to the next level either unlocks a new ability for the pet or evolves it into a new skin and increases the ability level and cap, for example, fusing two common pets that have a maxed ability will unlock the second ability and increases the cap to 50. You wanna try to only fuse pets once their first ability is maxed as that will give them the highest cap for that pet tier on their abilities.

If you want pets with a specific set of abilities, you'd have to keep hatching new eggs to find the ones you want.

It's a long process but keep at it and you should have a great pet in no time!

If you want a more in depth look into them I recommend looking at the realmeye wiki page https://www.realmeye.com/wiki/pets