By far the easiest and quickest way to reach Yakushima is by plane. JAL flies from Kagoshima five times a day and it takes only 35 mins. But you can also fly direct from Osaka (1 hour 30 mins) and from Fukuoka (1 hour) once a day. If you're coming from Tokyo, it's normally better to fly to Kagoshima and get a connecting flight or change to the jetfoil or ferry in Kagoshima city. Click the plane for more information.
There are 2 daily ferries running between Kagoshima and Yakushima. They both go from different ports so be careful.
Ferry Yakushima 2 leaves from the main port in Kagoshima city next to the jetfoil port and it takes 4 hours. The ship is large, has an outside deck and a carpeted area to stretch out and relax inside.
For more information about Yakushima - check out the one and only Yakushima: A Yakumonkey Guide