The (usual) first pet in the game is a Baby Bluebell, which are the blue puffball monsters you see in the Saliko Fields. After going to the map two maps right of Curly's farm, you will see an NPC named Theo, who begins offering you quests. Once the quests are completed, you will have to find a lost baby bluebell roughly four maps to the right of him, and guide it back to Theo. After completing the "Lost in the Forest" (I think) quest, you will unlock skills used to command your pet, and the "Call Pet" ability.
I hope this helped!
