To take care of your livestock and poultry you need to hire Gotz to build animal barns for them to live in. The barns will protect your animals from rain or snowy wheater. If you leave them outside in bad weather your animals will get wet and become sick. Gotz can build up to 7 animals barns on your farm, giving you 28 animals to take care of. What types of animals you want to use to fill those barns is up to you. You could fill your entire farm with chickens as long as you have 7 poultry barns.
The animals you purchase will have a natural life span even though your own character will live forever. The chickens and ducks will start to pass away at around 5 years of age, the sheep around 6 years, and cows will die after 7 years or so. You can check the age of your animals by going into your Farm Summary page and tapping on the animal heads to reach the overall animal summary screen. If you select a specific animal you can see it's age and current level.
Unfortunately when an animal dies, whether from natural causes or illness, you take a big hit on your Friendship Points with everyone in the valley. The best way to avoid this is to make a habit to save your game before going to bed, and if your animal dies the next day just reload your game until you finally reach a morning where it still lives. To sell the animal you will have to use your telephone to call Popuri (for poultry) or Rick (for livestock) and unload it for a profit before it dies.
Besides your own animals you will see various wild animals scattered around the Forget-Me-Not Valley. The wild animals are there just to add charm to the area. You can not own the animals as pets, and the animals do not indicate better fishing times like in A Wonderful Life.