The main airport in Shimane is the Izumo Airport. There are several flights every day going from Haneda to Izumo, and from there, you can take a bus from the airport to Matsue Station, in about 30 minutes. Overall, the trip takes about 2 hours and is the quickest way to get to Shimane from the Kanto Area.
By Train
Trains will take a bit longer due to the fact that there is no bullet train that goes from the major stations in Japan directly to Shimane. From the Kanto and Kansai regions, you will need to take the bullet train to Okayama, and from there, get on a special train known as the Super Yakumo Train to get to Matsue Station. Overall the trip will take 6 hours from Tokyo (40,000 yen both ways) and 3 and a half hours from Osaka (20,000 yen both ways).
By Bus
Bus is by far the cheapest way of getting to Shimane, and there are various overnight and regular highway buses going from the Kansai and Kanto area, costing around 23,000 yen both ways for Tokyo and 10,000 yen for Osaka. It definitely takes longer to get there, however, taking about 11 hours from Tokyo and 7 hours from Osaka.