Every Icelander grows up reading the sagas and any traveler seeking some understanding of Iceland and its people could not do better than to start there.  You can start with more modern works but since all Icelandic literature owes some debt to the sagas, you will need to turn to them eventually.

The sagas are part history, part fable.  Most of the authors have been lost to history (Snorri Sturluson may be the author of Egil's Saga) and the exact time of composition of the sagas in unknown.  Generally most seem to have been written around 1200-1350 about events happening between 850-1030.  The sagas are best understood within the context of other sagas so it's good to read more than one. Some of the most influential are Njal's Saga, Egil's Saga, and The Saga of the People of Laxardal.  Americans especially should be interested in the Vinland Sagas-- The Saga of the Greenlanders and Erik the Red's Saga, which tell the tale of the discovery of America (Vinland) by Leif Erikson, son of Erik the Red.

The elder or poetic Edda is a collection of anonymous Icelandic poems and writings.  The younger or prose Edda, written by Icelander Snorri Sturluson, seems to be some kind of instruction manual for up and coming poets, with lots of information on Norse myth.  Both are extremely important sources of information about Norse myth.

Besides Snorri Sturluson, there is probably no better loved Icelandic author than Halldor Laxness, winner of the Nobel Prize for literature 1955.  His masterpiece is  Independent People, about a stubborn (and often stupid) Icelandic farmer who goes through many struggles and hardships in his attempt to live independently.  But Laxness wrote many other novels as well, many available in translation.  Under the Glacier, for instance, is a delightfully bizarre story, packed full of eccentric characters living in a mysterious rural village.

For the slacker version of Iceland, especially its capital, look no further than 101 Reykjavik by Hallgrímur Helgason (also made into a movie). 

Further reading on celandic Travelogues and Essays:

  • Neither Here nor There: Travels in Europe by Bill Bryson
  • Naked Europe by Jon Breakfield
  • Iceland: Land of the Sagas by Jon Krakauer