Core Keeper

Pet are pretty rare but you can find them if you know where to look from. Every slime boss drop their respective slime pet. But the fastest and easiest way to find is, "owlux" from "lively egg" that have 1/25 chance of dropping from wooden creates. You find a lot of this create inside the stone and rock biome close to the core.

When you have your egg you need to put it inside an "egg incubator" that you can craft on a copper workbench. Then you need to place the egg inside and power the egg incubator whit electricity.

To make electricity you can make a lever or a generator just press A or right click on it after placing it down or it will not generate electricity . You know the egg incubator is powered when it glow in yellow a lot and the egg start to cook.

The description of the egg incubator say " dont let it at the wrong temperature" dont worry it's only for the lore. After 30 minute the egg will crack and you need to collect it just like the food that you can cook.
You then need to right click or use the egg in your hand to make your pet alive.

Every time you make an egg the color and special competence chance.