A Lot of places offer bikes for rent at a fair price.
Fanoe has a good amount of bike trails. You can ride on the well built, 2-track bike trails from Nordby to Sønderho.
Most bike trails outside of the villages are basically separated from the roads by a few meters of natural scenery.
So, go ahead, rent a bike and explore the Island or come down to the beach and visit the Kite Fliers!