Getting About
Finding your way around Rhodes is very easy as there are only a small number of main rhodes which pass by most areas of interest. I highly recommend hiring your own car to be able to fully appreciate the island and see the places of interest.
Car Rental
I've always reliably booked via www.economycarhire.com, getting to the villa from the airport is very simple taking just two roads and approx 30-40 mins depending on traffic.
Taxis
Taxis are plentiful and cheap. If you are averagely mobile on your feet it is a nice walk to walk into Lindos for dinner and then a simple taxi ride back (€4). Rhodes is very safe with virtually zero crime, so walking around at night isn't an issue.
Taxi from the airport is approx €50-60 (as a guide only)
Bus
I wouldn't recommend catching a bus from the airport, it would be a long journey via Rodos town involving a change.
For daily trips however the Buses during the day are plentiful and there's a bus stop at the bottom of the road with time tables on. They tend to meander around smaller towns and villages which can be a nice adventure.