After repairing the mill and completing Hayden's follow-up quest in Fields of Mistria, you can start buying chickens and cows. To obtain more animals such as ducks and sheep, you'll need to upgrade Hayden's farm. The most obvious way to get animals for your coops and barns is to adopt them from Hayden's shop. Another way is to breed them, and if you want to start unlocking different colored breeds, you'll want to take advantage of Mistria's animal breeding mechanic. However, it isn't enough to have one male and female of a species.