I looked up harrison's cave for you, and it looks spectacular. We are going to Barbados soon ourselves. But I am not sure if the cave is open, as they did close it for renovations. I believe it is open, as I see reviews posted from 2008, but it was closed for most of 2006, 2007.
as for taxi drivers, you can almost always be found at popular tourist places - plus almost every taxi driver I have ever had in the Caribbean has asked us if we want him to pick us up and they will come back for you at a set time. You can easily set this up with a driver before you go there, ask a few drivers if there are willing to take you and pick you up.
Plus - Barbados has a big "tourist" area at the port. I would bet you can find the tour locally when you get off the ship.