Your dog takes exactly two full seasons to grow, so it will be an adult dog by Fall 1 of the first year. As a puppy, you will not be able to do much with it aside picking it up to raise its affection. Once it's fully grown, there will be more mini-games and uses for your dog.
As an adult, you can play a frisbee mini-game with your dog at the beach. You will need to buy a frisbee from Won for this (it costs 5,000 G), and head down to the beach with your dog.
If you investigate the white sign next to Kai's shack, the game will ask you if you want to play frisbee with your dog. Choosing yes will prompt the mini-game to start.
The premise of the game is very simple: a bar will be on the left of the screen, with a red portion and a green portion to it. The red part indicates the 'out of range' zone, and the green part indicates the distance your dog can run. There will be a cursor that moves up and down the bar, and your job is to throw the frisbee (stop the cursor by pressing A at the right time) so that it falls within the distance your dog can run, but not so high in the red that your dog won't be able to catch the frisbee.
At first, the green bar will be significantly smaller than the red bar, as your dog won't be able to run as far. But as you 'practice' more (train more often), the green portion will increase slightly, and you will be able to throw a further distance for your dog to catch.
Every year, on the first day of Summer, there will be a frisbee throwing competition you can enter with your adult dog. Now, the more you have trained during the year, the further your throwing/catching range will be, and the higher your chance of winning will be. You can win a Power Berry from this event, if your record was the highest.
Your dog, when full grown, can also chase off wild dogs that may come into your farm and scare your livestock at night, if you keep them outdoors. A solution to this is to put your own dog outside, and it will be able to chase off the wild dogs.
Showing your dog to the various residents of town will give you a wide range of responses. The majority of the townspeople generally adore and like dogs, but some will react in fear or disgust, so be careful, if you don't want to lose any friendship points.