1. Sports Authority
Probably the largest supplier of athletic supplies in Japan, including weight training equipment, camping gear, and a full selection of shoes. Usually, Sports Authority stores are located in the biggest shopping centers: AEON in south Kagoshima, Canal City in Fukuoka, Diamond City in Hiroshima.
2. Chuo Sports (中央スポーツ)
Rather than taking the special AEON bus from Kagoshima central station (鹿児島中央駅) for thirty minutes, check out Chuo Sports, located just west of Tenmonkan (天文館). Although it clearly doesn't offer as much variety as major superstores, you can find baseball equipment, running shoes, and training clothes, as well as a section set aside for women.
3. Hayakawa Sports
Hayakawa would be my recommendation for running shoes and apparel in the Kagoshima area; the people there know about the distance, about the speed, and stock their shelves accordingly. A full assortment of racing spikes, distance shoes, singlets, and decent shorts. Local race information is also available. In Tenmonkan, just a few minutes' walk from Starbucks.