While on a walk, pass all the "?"s in the corners on the screen. The farther the item is away from your house, the higher chance that it will be something rare.
To have your dog run away and bring you back a rare present, first get your dog in very good condition. Then, leave your game on without playing it for about one hour or until your dog runs away and brings you back a present.
Set the system date to your birthday, then go out for a walk and try to go through all of the "?" marks that are near your house.
To get better presents, get your dog in a good mood by feeding, washing, brushing, or playing with him or her.