As soon as the game starts, you will find a stray dog on your farm. There are two types of dogs that come around looking for food: the typical floppy-eared dog with a red scarf, and a dog that has pointy ears with a blue scarf.
At first you cannot interact with any dog, however, there is a blue food bowl outside of your farmhouse where you can add food on a daily basis. The food that you put into the dog bowl can be any edible item, but you can gain Friendship Points faster if you add fish or milk on a regular basis. Feeding the dog every single day will allow you to eventually adopt the dog of your choice. In order to "pick up" the dog, you will need a total of 31 Friendship Points. Make sure that you pick up the dog that you want, as you will not be able to change. After you have chosen your dog, caring for it is one of the easiest tasks in the game: pick up your dog once and feed him once a day. It is recommended you wait until Spring's rainy season has passed. This will allow you time to collect money and lumber to build a dog house for your furry friend to live in when it rains.
The higher your dog's Friendship Points, the easier it will be for you to train him to find Power Berries. There are a total of 5 Power Berries to be found in the game. Each Power Berry will raise your Stamina by 20 points, and raise your Fatigue Resistance by 10 points. You can train your dog by buying an Ocarina from Louis' Item Shop for 450G. To use the Ocarina, you must equip it from your Tools menu, and while holding the "square" button on your PSP you'll use the D-pad to play a tune.