In HM: Island, there are the basic animals: Dog, Horse, Chickens, Cows, and Sheep.
Obtaining a dog in this game is a bit more complicated than usual. First, you must befriend Mirabelle to two hearts. Then, after saving up sufficient money, you must have Gannon build the second bridge. After those requirements have been done, walk over the second bridge, into the forest where you will meet the dog.
In order to get a horse, you must befriend Charlie and Chen to two hearts each, then visit them and they will give the horse to you. You can ride the horse around the island, which will help quicken the time it takes to move from place to place.
Before you can buy Chickens, you must have Gannon build the Chicken coop on your farm. Similar to most other games, chickens require the least amount of work to take care of. They're kept in the chicken coop, and lay eggs that can be sold.
Chickens must be fed chicken feed you buy from Mirabelle. If you leave your chickens outside during nice weather, they do not need to be fed.
Cows are typically the most productive animal in a Harvest Moon game. You can buy cows from Mirabelle, after Gannon has built the barn on your farm. Brush, milk, feed, and talk to your cows daily.
Sheep are very difficult to get, and will most likely be the last animal you buy. The requirements to get sheep are to have Mirabelle at three hearts, own four cows, and own four chickens. Sheep are taken care of the same way cows are. Feed, brush and talk to them daily.
* May vary pending animal's affection level