Pretty simple, this is. First you must have your pet fight alongside you. How do you do that you ask? On the Minions > Manage screen, select the pet and click on Start Following and Turn Combat On. This way your pet follows you everywhere, and it fights with you when you enter a fight.
Now after a few fights, your pet should have enough Xp to level up. To go back on the Minions > Manage screen and click on the pet. Look at it’s Xp, and if the first number exceeds the second (I.E: 4900/1000) then your pet can level up. It’ll subtract the second number from the first. (I.E: 4900-1000=3900) and the remainder number is how much Xp your pet will have left.
If the pet you are using to fight with you is your mount, then you’ll have to go to Minions > Mounts to level it up. Other than that, you’re perfectly fine.