In order to trigger this quest you need to complete Getting Started, one of the first quests in the game. Once you’ve completed the basics of the basics - namely creating some crops and harvesting them - this quest will pop up.
Raising Animals is a bit expensive for an early quest, as it requires you to construct a building that can house animals. You can create either a Coop or a Barn to fulfill the requirements, the former of which houses Chickens and Ducks, the latter of which houses larger animals like Cows. You’ll need 4,000 gold, 300 Wood, and 100 Stone for the Coop, and 6,000 gold, 350 Wood, and 150 Stone for the Barn. I recommend starting with the cheaper, easier-to-maintain Coop, but it’s your call.
Both buildings are constructed by Robin. She’s located at the Carpenter’s Shop, located in the north of Pelican Town, and her store is open from 9 am to 5 pm on most days. (I went on a Friday and it was open, if that helps.) Be sure to clear a spot for her to work on your property.
Robin will require several days of work to finish the job, and she won’t work on days she wouldn’t normally be at her store. Once either the Coop or the Barn is complete you can snag 100 gold from the Journal to end the quest. And, no, you don’t actually need any animals… though if you’ve come this far, why not go a little further…?