Nonstop flights on Icelandair [http://www.icelandair.net] are available at the best value from the U.S. and Canada, with gateways in Boston, New York City, Minneapolis/St. Paul, Orlando Sanford, Halifax, and Toronto. Destinations beyond Iceland include most major European cities (i.e. Amsterdam, Berlin, Copenhagen, Frankfurt, Glasgow, Helsinki, London, Oslo, Madrid, Manchester, Milan, Munich, Paris, and Stockholm, with newly-added cities Bergen and Gothenburg), with Icelandair's hub-and-spoke network connecting via Keflavik in Iceland. (Please note that some destinations are seasonal.) You can also stopover in Iceland for up to seven nights at no additional airfare on your way to or from Europe. That's two destinations for the price of one!
Once arrived, take the Flybus, it gets you Hafnarfjordur several hostels and hotels. Hafnarfjordur is also the closest part of the Greater-Reykjavik area to the Leifur Eiriksson Airport so it easy to take the Taxi.
From Reykjavik it is easy to take the public bus that runs every 15-20 minutes.