Tanegashima has two claims to fame. In 1543, Portuguese explorer Mendes Pinto found his way there and introduces firearms to the Japanese, and for a long time thereafter rifles were known as tanegashima in Japanese. These days, though, the main attraction is Tanegashima Space Center, JAXA's main spaceport.
New Tanegashima Airport (IATA: TNE) has five flights daily from Kagoshima (30 min) and one flight daily from Osaka-Itami (1:20), all on Japan Air Commuter , a JAL subsidiary. The airport has scheduled bus services to Nishiomote (38 min, ¥790) and Iwasaki Hotel (62 min, ¥1270). A taxi to Nishiomote costs about ¥6000.
Several operators have services from Kagoshima, with some continuing onto Yakushima or Kuchinoerabujima. The largest is Cosmo Line, whose Rocket fast ferries run four times daily (1.5 hours, ¥4,700) with one service from Tanegashima onto Yakushima, while the Princess Wakasa car ferry runs once daily (3.5 hours, ¥3,300). There is a slight discount for purchasing a round-trip ticket with the same company, even if you choose to take the Rocket one way and the Princess Wakasa the other.
The easiest way to get around Tanegashima is to bring or rent a car. Contact information for taxi and rental car companies is here: 
The Daiwa Bus service ended in February, but there is a new community bus service. The schedule and route for Minamitane (South Tanegashima) is here:  Information on bus service in Nishinoomote and Nakatane should be available soon.
Tanegashima Space Center, ☎ . Tue-Sun 9:30 AM-5:30 PM. Japan's largest spaceport. Free exhibits and guided tours three times daily (reservations required). No entry during launch days, but there are designated viewing points around the perimeter. Free.
Chikura Caverns (千座の岩屋). White sand beach with caves that can be entered at low tide. There are picnic tables at the edge of the beach. There are a couple food stands and drink machines. free.
Takezaki Beach (竹崎海岸). Beach at the Tanegashima Space Center. Long, white sand beach with areas for snorkeling close to the east side. Restroom with changing area and place to rinse feet is available beside the free parking area. Covered picnic tables nearby on lawn of space center. free.
Cape Kadokura (門倉岬). Cape overlooking the rocks and shore where the first firearms entered Japan. Beautiful view. There is a lookout post shaped like an old ship. A small shrine is nearby. Restrooms available beside the free parking area. free.
Tanegashima is a well known destination for surfers in Japan.
Moonbar  The best place. Breyen-san and Tomoko-san is running the cosy place perfectly.
- Yakushima, mysterious island of giant cedars