Fanø is in Denmark.
Fanø is one of the Danish Wadden Sea Islands.
You get on the island with the ferry from Esbjerg. Travel time is 12 minutes, a return ticket for adults costs 37 DKK (~10 EUR). Schedules available at http://www.scandlines.de/en/main.htm (select the route "Esbjerg - Fanø"). The ferry arrives at the city of Nordby.
Travelers arriving by train in Esbjerg can take the bus line #5 from the main station to the Fanø ferry port. The ride takes aprox. 10 minutes and costs 8 DKK (~1 EUR) for adults.
Most convenient way to travel on the island is by car or bicycle. Cycleways are excellent. There is a bus line going from Nordby to Sønderho.
The beach, the town of Nordby, the town of Sønderho.
Rent a house (for example through http://www.danibo.dk), stay at the Vesterhavet hotel or at the Fanø Badeland hotel (http://www.strandhotellerne.dk/Default.aspx?AreaID=73)
Take the ferry, return rides are free (you have paid for it when you bought you ticket in Esbjerg).