If you do not want to rent a car during your stay in Fuerteventura, then maybe the 2nd best option is a Taxi.
You can start using it from the airport, instead of hiring a private shuttle service (known as "transfer"). Its price would be about 45 euros each way. Taxis are always available at the airport, no need to book in advance and is an absolutely safe service. If you are a larger group, just ask them to get a bigger one for you. Click on the pictures for more information.
Another alternative to the taxis is to contract a private transfer with a company. Prices and terms will depend if you hire an exclusive transfer or shared with other travellers. Here are some companies that provide this service: