You can build up your dog's affection for you by simply picking it up every day or petting it with the touch panel glove. Your dog can be a useful tool for raising people's LP/FP points, from the showing your pet trick.
You can also train your 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.