To get a pet you need to hunt for Atingo. He is around 75lvl and spawns in the Tower of Insolence, Orc Barracks, Dragon Valley and Plains of the Lizardmans.
Atingo can spawn at random places in the mentioned locations.
Once Atingo HP drops below 50%, there is a chance for a pet to appear (Buffalo, Strider, Tiger, Kookaburra, Hawk, Wolf). You need to kill Atingo first to make the pet ownerless. If you kill the pet, Pet Necklace used to summon the pet will be dropped with 100% chance.
There is also very high chance for Sin Eater to appear, then no pet will be dropped.
Pets do not steal XP from Pet Owner, but they get their own XP based on how much DMG to mosnter they did during fight.
Pets can have some offensive, defensive and self-buff skills depending on pet type. You can use pet for some additional DMG, but after 60lvl of the pet, you will need to spend high amount of adena for Pet Food.
https://www.lineage2.com/news/pet-system-preview