Leave your hotel around 9AM.
The first stop today is Hagi, an old castle town in Yamaguchi, once the capital of the Mori Clan, a powerful samurai clan in the feudal period. These days it is a very well preserved castle town with numerous residences that can be visited still intact. It is also famous these days for its local pottery.
Next, a trip to one of the most important shrines in Japan, the truly astounding (and huge!) Izumo Taisha shrine in Shimane, which is said to be the oldest shrine in Japan, and arguably its most important.
Stay overnight in Tamatsukuri onsen, a charming riverside hot spring town in Matsue, the streets of which are lined with interesting statues from Japanese mythology.