Omi Shrine is a holy place where religious souls meet to pray. You can do the same, or respectfully observe others in worship.
You'll be spoiled for sightseeing choice in Hyuga. Omi Shrine is a good place to start here, but there are loads of other fine places to experience. Okuragahama Beach, and Hichiya Castle are not far from the sacred place and and will cap off your day nicely.