When you begin the game, you'll notice several stray dogs wandering around. If you approach them, they'll run away. To woo a dog to your yard, keep putting food in his bowl. Anything will do: grass, eggs, berries. When you place these foods in the bowl, they immediately turn into doggie mush. After a while, you will notice the dog appearing at your farm more and more. Keep trying to pick up the dog (X). At some point, he will let you pick him up, and he will be yours!