The trip to the Jaxa Space Center needs to be self-organised. Please make sure that you double check times, since this information might be outdated.
It is possible to get to Tanegashima with a ferry from Kagoshima to the city of Nishino-omote, from there is a hour drive. A day-trip from Kagoshima is possible, but will be a rushed visit. For that you need to take a morning ferry (I would recommend 7:45 and take the 17:05 ferry back to Kagoshima). I would recommend staying another day to explore this beautiful island.
The "Toppy" (トッピー) and "Rocket" (ロケット) Jetfoil ferries can be taken from Kagoshima.
Timetable in Japanese (google translate will be useful)
Pier in Kagoshima:
If you have some waiting time, you can get some coffee or food in this nice cozy cafe: Cafe web site / visitor's review
Best (and possibly only) way to get there is renting a car:
They have a shuttle from the ferry to and from the Rental car offices and the ferry pier. The trip from Nishino-omote city to JAXA Tanegashima Space Center takes around 1 hour: Google Maps
More information about the JAXA Tanegashima Space Center
There is a regular bus service, but it is very risky to rely only on it
because the ferry schedule is easily affected by the weather,
the number of bus and ferry service is very limited, and if the connection is not as schduled,
there is no other public service to back up at all.
Taking it into account, the only time schedule by using the regular bus service is as follows. Of course, we cannot guarantee the schedule at all.
I stayed in the Tanegashima Araki Hotel, which also provides access to the Onsen next door: Rakuten Travel Site
Map with additional information about Tanegashima island: Tanegashima Tourism Map (PDF 2 pages)