Pets perform their various tasks within a certain limit that can be raised by leveling them up.
In order to level up a pet, you must collect XP (Experience Point) Potions. Your collected XP can be seen in the bottom right corner of the Pet menu. While looking at a pet, pressing the XP button will give it a certain percentage of the total XP in your collection – this can be seen at the top of the screen.
When you have enough experience points added, your pet will automatically level up and increase the amount of help they give you. Each pet has his own XP bar and is leveled up separately. XP is also be earned each time your Pet performs their action and they may level up due to this as well.
Please read this article for information on how to acquire XP Potions.