Here you can read about them and also find out which option suits you best. Whether it be the cheapest, most convenient, or most adventurous one.
By Plane
You can fly direct from, Osaka, Fukuoka or Kagoshima. Otherwise you would need to change flights at Kagoshima Airport.
From Osaka Itami Airport - Yakushima Airport (90 min)
One way fare is about *¥34.000 = 280€ ($305),
One way fare about ¥24.000 = 197€ ($215)
One way fare about ¥10.000 = 82€ ($90)
*please note that the exchange rates can fluctuate and this is only a guideline to help you.
By Ferry
This is the fastest and most popular way, also there are about 8 round trips per day. The one way ride takes about 2 hours depending whether the ferry makes a stop at Takushima or not. The price for a one way ticket is 8.400 = 69€ ($75), so this ferry is the most expensive one of the three listed. Furthermore you should remain seated and are not allowed to go on deck.
March 11th 2017 to Sep 18th, 2017 Summer timetable

Sept 19th 2017 to Nov 30th, 2017 Autumn timetable

Dec 1st 2016 to March 10th, 2017 Winter Timetable

Leaving every morning from Kagoshima at 8:30 to Miyanoura Port and returning in the afternoon at 13:30 , the one way ticket price for this ferry is ¥ 4.900 = 40€ ($44). If you wanna bring your car with you the price is about ¥33.000 = 271€ ($296) there and return. Estimated travel time of this ferry is about 4.5 hours.
This one is the most adventurous and cheapest, as you are travelling with an overnight freight ferry. Leaving at 18:00 and arriving at 7:00 at Miyanoura port, returning to Kagoshima at 8:10 and arriving at 14.00.
Yes 13h, but it’s overnight so you sleep most of the time and you can spare yourself an accommodation for one night.
You are allowed to walk around and get on deck which is pretty nice.

But now...
Where do I get the ferry tickets?
Toppy or Rocket hydrofoil:Online
At the port where the ship is leaving.
How do I get to the Port?
If you want to go by Train, take the Kagoshima Tramway for Kagoshima eki mae. Get off at a station near the port, for example Asahidori.