If they are an Eggling or Hatching you can nurture them.
For Adolesants or adults it is best to pair them with a strong sylesti in a weak place, or send them on a level 0 mission. Don't forget to equip them if you can. (I can't guarantee they will succeed on the mission. It depends on their stats.)

@Krinadon, This should probably be in FAQs

@Calypso1,

After you have created your pet, you have to add it to your party via it's profile page. There should be a 'Add to party' button which you should click. If not, click on 'disband', then click on 'add to party' again.

Afterwards, you will find that a location on the map is unlocked, which you have to follow the tutorials to get the quest. After doing so, go to the next area designated on the quest log, and battle other mobs. After each battle, your pet (in your party) will gain a certain amount of exp. You will find that after accumulating the target exp, your pet will level up.

It is very important that you follow the tutorial to progress through the stage. It will help you understand the basics of Sylestia, and if you have any extra questions, feel free to ask them in the Newbie FAQ section of the forum.

Hope this helps

@Calypso,

Moved to FAQ Forum.