The Izumo-Taisha Grand Shrine is about 20 minutes from the train station. The current complex was rebuilt in the 17-8th centuries and attracts visitors for its grand straw festoon and architectural style.
Cleanse your soul before you enter the sacred grounds.
The sanctuary is actually closed to the public but you can peek from the entrance to see the grand scale, then walk around its walls.
The star attraction is the 13 m long, 9 m wide sacred straw festoon that weighs 5 tons.
The Ichibata Electric Railway takes a scenic route towards Matsue in about an hour.
Matsue Castle was originally built in 1611. To encourage foreign visitors, show your passport and get half off admission all over town. The castle isn't hard to access with a fairly easy climb.
Behind the castle is a historic district with several samurai homes and museums. Having experienced some at Bitchu Takahashi already, I didn't find a significant difference.