Harvest Moon: Save the Homeland

To first obtain a dog:
You'll find that there's a food bowl beside your house. In order to entice one of the two wild dogs to be yours, you must first befriend them. Drop something into the bowl to turn it into dog food, and the best part is that it doesn't even have to be edible. For this reason, you can pick up wild flowers or grass or herbs (probably red herbs, not the regular herbs as you do need those for an ending quest) and use them.
After several days of leaving food out for the dogs (I'd safely bet on two weeks being enough), you should be able to chase after the dogs and pick it up. Once you pick it up, you are given the chance to name it and claim it as your own!

To care for your dog:
-) Bring it inside before it rains.
-) Don't pick it up often if it's sleeping (you can pick it up once if it falls asleep while putting away other animals and you're heading in for the night).
-) Always give it food every day, even if it's not outside to eat it.
-) Befriend Louis and one day, during the afternoon, you should get a scene from him at Maple Lake. He will give you the flute, which allows you to train your dog to do specific tricks.
-) Pick your dog up at least once a day, as that's your way of giving it love.
-) You can leave your dog outside when you buy a doghouse, but it's better to just bring him in anyway before it rains.

For me, I received this trophy within a year's time. My game lapsed halfway through the first half of the year, so come Summer 4, after spending three and a half seasons taking care of it, when I went to pick it up to bring it inside, the trophy was given to me.