Fanø is in Denmark.
Fanø is one of the Danish Wadden Sea Islands.
You get to the island by ferry from Esbjerg. Travel time is 12 minutes, and a return ticket for adults costs 40 DKK (~5 EUR). Schedules available at http://www.faergen.dk/ (select the route "Esbjerg - Fanø"). The ferry arrives at the city of Nordby.
Travelers arriving by train to Esbjerg can take the bus line #5 from the main station to the Fanø ferry port. The ride takes aprox. 10 minutes and costs 20 DKK (~3 EUR) for adults.
The most convenient way to travel on the island is by car or bicycle. The island is connected by a main road from Nordby in the north and Sønderho in the south. The 2 km wide beach serves as a road year round connecting the beach town of Fanø Bad to Sønderho. There is a bus line going from Nordby to Sønderho. In the summer cars drive on the beach all the way to the shore.
Cycle paths on the island run along the main roads, and all other roads are safe to bike.
The beach, the town of Nordby, the town of Sønderho.
There are multiple options for staying the night on Fanø. Most tourist rent a summer house for one or several weeks:
Take the ferry, return rides are free (you have paid for it when you bought you ticket in Esbjerg).