The only scheduled service to the island is with the RMS St. Helena departing from Cape Town and Southampton. It is the only remaining British mail ship. Schedules and tariffs one can obtain from the web page address below.
There are buses that service the people of the island - they are not really tourist busses as such. But nothing stops a tourist from catching one of these busses to get to a destination.
You can hire a motor car with a driver for a day. I would not recommend driving yourself. These roads are different from roads in the rest of the world. If you don't know at which bend to stop and hoot before continuing or at which bend to stop and reverse, you may land in the sea or down the bottom of a hill!
You could also hire a metered taxi to get from one place to another.