Fuerteventura with its total area of 1660 square kilometres is double the size of my hometown Berlin but it only has 110.000 inhabitants. Even if you include the tourists you'll always have the possibility to be alone if you want. So there's more than enough to discover.
I hope my geographically overview can help you to decide for a round trip or a special destination depending on where your preferences are.
Morro Jable und Jandía
white beaches and tourist
Punta de Jandía / Cofete
wild "jeep bus" ride
Faro de la Entallada
lighthouse on a mountain
Costa Calma
beaches and karoke