Fuerteventura BusesThe Bus company, Tiadhe, provides a fairly comprehensive, reliable and economical service for the Island. While you will not be able to go everywhere on the island using public transport (really you would need a 4x4 for that), it does offer a means to explore much of the island without hiring a car.
If you are thinking of using the Bus (or Guagua as they are called here) to get from the Airport to your accommodation, remember that most routes do not stop at the Airport (strange that) and you will need to get to Puerto del Rosario first by either Taxi or Bus. This option may well be worthwhile if travelling in alone or in a small party as Taxi fares can be expensive over long distances.
Tiadhe Website of the Fuerteventura Bus Service. Click on Rutas/Horarios to view Routes, Fares and Timetables.