Getting between Hagi and Tsuwano there are basically two main transportation options.
Bocho bus operates a service between the two towns. When I took it, the trip came in at around one and three quarter hours. The bus was fairly empty. The price was 2,190 yen one way for an adult, which I feel is a bit steep. You pay on the bus and change is available.
The bus originates at Higashi-Hagi station, stops at Hagi bus station and terminates outside the JR station in Tsuwano.
The timetable can be seen online (in Japanese, but if you know the characters for the two towns it is easy enough to understand) here.
The train is a cheaper but slower option between Hagi and Tsuwano.
A connection is required in Masuda, which means that the total journey time is somewhere north of two and a half hours – depending on the time of day, it may be closer to four hours.
The one way adult fare is 1,660 yen.