Harvest Moon: Save the Homeland

How to obtain a horse:
Befriend Bob. Bob is the owner of the only ranch in the homeland and also operates the supply store where you can buy cows, chickens, fodder, feed, and medicine. If you give him several eggs, this will raise his compatibility with you. Furthermore, choose to work part-time for him and his ranch. You will be given a temporary set of a brush and milker to tend to his horses and cow. Make sure you put food in the feeder in the barn, and milk, brush, and talk to the cow. When it comes to the horses, choose the type of horse you wish to have. Make sure you talk and brush to that one while brushing all of the others. The more compatibility you have with a specific horse, the more likely you'll end up receiving that one.

After receiving several eggs and working part-time enough, one morning Bob will come by and deliver the horse, asking if you'd help raise it for him. You may name it, and after this cutscene, you will find it inside your barn. Make sure you care for it as you would any of your other pets.

How to ride your horse:
Upon receiving your horse, it will have no further love for you. The progress you put towards it from the ranch is reset, so you have to actually give it some attention. Brush and talk to it (from the front) for a couple of days (it was exactly three days for me) and you will eventually receive your first heart. This is all you need in order to mount the horse. Approach and interact with it from the side to climb on its back and receive this trophy.