On your farm you will have 3 animals that are there for companionship. You can't sell these guys for profit, but they do add a little bit of charm to your farm.
When you first start your game you will be given a dog and a cat. Both pets will be at their full grown size. The cat doesn't do anything useful on your farm, but you can train the dog to chase the wild dog that might appear on your farm. At 6pm you will be notified by Mars, the Channel 4 Sprite, if a wild dog appears on your farm. The wild dog can bark at the livestock and poultry that you have left unfenced outside, scaring them and making them grumpy the next day. If you have your dog outside it may chase the wild dog around and keep it occupied so it doesn't get the chance to chase your chickens.
You can train your dog to chase the wild dog by playing ball with it. Thomas will give you a Dog Ball (for free). If you throw the Dog Ball near your dog, it will retrieve it for you. Keep throwing the Dog Ball around to train your dog. You can see how well your training is going by looking at your dog's statistics from inside the Farm Summary screen. If you lose (or give away) your Dog Ball then Thomas will appear again and give you a new one.
The other pet you can receive is an adult horse. After you have shipped 1000 items via your Shipping Bin, Takakura will stop by in the morning and deliver your horse. You can use that horse as a portable Shipping Bin, and even ride it around the valley.
The pets can also be rubbed using the Touch Panel Glove. You can rub all three pets and you can wash your horse if you have the Brush tool equipped along with the Touch Panel Glove accessory.