I don’t know how many days you have, so I’ve just been thinking of a 3-day trip to Shimane. Of course, more (or less) time could be given if you have different plans.
The first day would include your arrival via bus from Hiroshima, both the easiest and cheapest way, and discovering Izumo city. Izumo is quite small and its main sites are nearby, so even if you arrive after lunch there is plenty time to see the city and enjoy it well. The shrine is much quieter after sunset, once the tour buses leave, and in my opinion that is the time it’s at its best. There are a few ryokans as well as cheaper inns and I think it’s a nice city to stay overnight.
The next two days would be for visiting some of Shimane’s other best sites, from old towns in the mountains, traditional performances, gardens, hot springs, a castle, boat ride, temples and shrines… Depending on your dates, you might be lucky and coincide with a festival too.
You’d return to Hiroshima on your fourth day, perhaps with an afternoon visit to the peace sites around town.
Of course, this is just a first draft without knowing much about your group and interests, but if you’d be more interested in other things we could easily change them around.
Here are some photos from my own travels around the area, featuring some sites that would be included in the itinerary: