About the Work
With one Greek epic under his or her belt, the student by now doubtless has a few opinions about the genre, good, bad or indifferent. While the task of reading another of nearly equal length may no longer seem as daunting, a less than enthusiastic student may be asking why two are necessary.
The question of whether the two Homeric epics were actually authored by the same person (or indeed, were primarily authored by one person at all) is an ancient one to which I have nothing new to add, however I will venture to say (having read each several times now in its entirety) that there are substantial enough differences between the two to make the experience of reading them rather different.
Whereas the elements that strike one most in the Iliad are:
a) the at times intricate similes and digressions
b) the occasional moments of character complexity, motivation and history inserted into an otherwise "primitive" narrative and
c) the many ways one's soul may be sent howling into the underworld by the work of cold bronze
The Odyssey is a work with a different narrative tone, though clearly taking place in much the same world. The first thing the reader will notice is its structure. While the Iliad covers a linear (though briefly selected) period in the Trojan War, the Odyssey starts near the end of the ten years of wandering it chronicles and tells the intervening story through a series of flashback narratives from the mouths of different characters. While the Iliad changes place, character and tone in a way which seems out of keeping with the antiquity of the work, the Odyssey seems downright novelistic in structure.
Though the various incidents on Odysseus' trip home are the elements most often retold, these actually take up only a few chapters worth of second hand narrative in the original epic. At root, this is the story of a household and kingdom which are put out of balance by the loss of their head, and the efforts of the head to return home. It also sees, in a concluding book which some have accused of being a later interpolation, the introduction of justice and compromise into the epic world -- topics that will be much more key in the Greek tragedies which were written a few hundred years later.
Whether the Odyssey is the work of a later poet or school of poets in the Homeric tradition, or both epics are the work of a single man, we begin to see the importance of the city state and civil order over the heroic egos that drove the plot and values of the Iliad.
Selecting an Edition
Unless you are seriously dissatisfied with the translation you selected for the Illiad, you'll doubtless want to use the same translator for the Odyssey. Just in case you didn't pick up both books at the same time, and you want to fling us a couple cents via Amazon, here are the links to the editions of the Odyssey by the same translators mentioned before:
The Odyssey of Homer, translated by Richard Lattimore
The Odyssey, translated by Robert Fitzgerald
The Odyssey, translated by Robert Fagles
Odyssey, translated by Stanley Lombardo