Travel to Shimane

Hi,
You can do either. You might check rail schedules on Hyperdia. Solely using the JR Pass, you'd clearly get your money's worth for a 7 day and almost 14 day pass with just a round trip. But at minimum it would take you about 9 hours on the train to Matsue, for example, with at least 3 transfers. If you flew, you could be there in less than 90 minutes to Izumo Airport, and JAL has its Yokoso fare or Welcome to Japan Fare. That said, if you wanted to do a lot of traveling from there, you'd need to pay more. There is a JR Sanin Okayama Area Pass that might save you some money in that case.
It's impossible to give more info without knowing your plans more. But I hope this helps you make the best decision for you.

Best of luck.