Take the Flybus from the airport at Keflavik to Reykjavik and stay in the downtown hostel.
Next day, look for the smallest bicycle shop in the world. If you are not sure whether the shop you are in is small enough, keep looking. There is a smaller one. This is the one you want.
Announce to the owner that you wish to purchase a used bike to get around Western Iceland, with the idea being that you would sell it to another traveler at the end of your journey. He will tell you this is not possible. Thank him, and leave. It is very important that you do not dispute the bike shop owner on this point. Thank him, and leave. Crucial.
Return to the shop the next day. He will have somehow “procured” a used bike that is barely suitable for riding on a bike path, much less cross-country.
Go to Thingvellir, Geysir, and Snaefellnesness. Take dips in hot pools wherever possible. Use the cycle to cross the Kjolur Route through the Central Highlands. On the way, a lanky Pole with a runny nose will point the way on your map. This is Polski. He has been helping travelers on the Kjolur Route since the Early Age, roughly 980 - 1140 AD.
At Hrerravellir, enjoy the hot pool. Meet young magi from CERN as you reposition the hot water hose.
Upon your return to Reykjavik, set up your bike opposite a jewel shop on the main commercial street. Place a sign on the bike offering it for sale for kr175,000. Within five minutes, the jeweler will emerge and purchase your bike for that exact amount.
Thus you will have re-enacted the Saga of Vilaf, of the Hill People. You may return to your home lands triumphant. A large feast will be assembled in your honor, and you will be invited to blow the horn of an ox 19 times.