I came by car, so I cannot say anything about trains and busses. But Charlotta (@sjalen) has mentioned how to get there by train and bus in her Eksjö page. The train station is close to the town centre (approx. 200 – 300 m by foot).
Eksjö is located at the road 32 (from Motala at the northeastern part of Vättern Lake to Vetlanda, north of Växjö and the Kingdom of Glass). Leave the road at the exit direction Vimmerby (road 33). It will lead past the train station.
Transport to Skurugata Canyon:
there is no regular bus (I have asked in the tourist office), so the only way to get there is by car or hitchhike. It is approx. 20 km to drive, and there is no other path which is quicker on foot or bike. After the train station, drive north at the roundabout (into road Regementsgatan) and turn north into Ydrevägen. This turn is already marked “Skurugata”. Follow the signs from there.
For a better overview, I have added a screenshot of google maps.
Eksjö is off the main railway line but if you change in Nässjö (Stockholm to Malmö line) you will have no problem reaching it by a short motor train ride towards Hultsfred.
County buses run from Jönköping and Nässjö as well and there are Svenska Buss coaches from Jönköping, Linköping, Västervik, Stockholm and other places.