Enter Domina. Go to the point on the far left of the Domina Map again. Take the upper left path and you will meet Duelle the onion guy.
Listen to his directions and catch the monster egg running around the area. After you catch the egg the Mail Pelican will take it back to your Monster Corral in your back yard. Listen to Duelle’s instructions to learn how to raise pet monsters.
You need to wait a few Game Day’s before a monster egg will hatch. To speed up this process, just go to the World Map and keep running back and forth(move your character to different lands). Every land that you cross causes one day to pass.
Upon Completion: Warning: this quest is missable in the cactus diaries! Return home, talk to Lil' Cactus, go downstairs (watch him write the Event on the leaf), and go back upstairs to save your game before continuing.