There several airlines that fly to Puerto Del Rosario which is the capital of Fuerteventura and where the airport is located these being -
http://www.ryanair.com
http://www.flythomascook.com
http://www.thomsonfly.com
A great site for finding flights is - http://www.skyscanner.net.

Car hire can be obtained from about £80 per week depending on size of car from - http://www.autoreisen.es/

Local buses run regularly between the airport and the town, approx 10 euros
There are always local taxis available - fare should be no more than 50 euros.