The Epic of Gilgamesh is a Sumerian epic poem that dates back to the 3rd millenium B.C. and is the first piece of written literature in the world. It was enormously popular in ancient Mesopotamia over centuries and knowledge of it extended beyond to western Asia and very possibly to the Greek world (suggested by some similarities to Achilles in the Iliad and to Greek flood stories). The poem in the form you are reading it is actually a composite of a number of Gilgamesh stories from clay tablets found at a a variety of sites in Mesopotamia (see the Introduction in the Penguin).
In the story, Gilgamesh is presented as part god, part man, that is a legendary king from the distant past. But we know of a historical Gilgamesh of Uruk (from the Sumerian King List), who reigned ca. 2700 B.C. and it may be that legendary material was "grafted" onto this king. The poem contains both fictional aspects characteristic of heroic epic, but also reflects historical aspects. Since it was so popular, it is valuable to the historian of Mesopotamian culture because it reveals much about the conceptual and religious world, e.g., attitudes toward the gods, how a hero was defined and regarded, views about death, and friendship. It can also tell us something about political and social organization in a Mesopotamian city like Uruk, as well as its physical layout. For example, the prologue (p. 61) reveals that Uruk was a developed city to the extent of having fortification walls (note that it specifies the building material) and temples, that it was a polytheistic (many gods) society from the number of gods named. Page 62 makes it clear that Gilgamesh was ruler, that is, that there was a king of some sort, and it reveals something about the organization of society (it speaks of warriors, and nobles). These are the sorts of things that you should be trying to pick out as you read.
You are not required to read the Introduction, but, hey, you bought the book so you might as well get your money's worth, and it contains interesting and fascinating information about the poem, its discovery only in the last century, and its historical context, so it will help you to get the most out of the poem itself.
Also, note that there is a Glossary in the back of the book that will help you keep straight the various characters and give you ideas for names of your pets.
As you read, think about the following questions and then what your answers suggest about the society, its composition, its attitudes and concerns.
1. How is a king supposed to behave? What is appropriate and inappropriate?
2. Are there checks on his power in the city? What are they and how significant are they? How would you characterize the political organization of Uruk?
3. On the basis of the presentation of Gilgamesh, how would you define a hero in Sumerian culture? Is Gilgamesh excessive in his behavior, or is he only acting as would be expected of a hero (and a king)?
4. How important is reputation?
5. Why is it important for Gilgamesh and Enkidu to kill Humbaba?
6. What function do women play in the story?
7. How are the gods viewed? What is their function? How directly do they intervene in human affairs?
8. Is the world depicted a safe and harmonious place, or precarious and uncertain? What are the problems one could face? What views about nature emerge from the story?
9. How would you define the relationship between Gilgamesh and Enkidu? What is important about friendship to them?
10. Why does the flood happen? Why is Utnapishtim singled out to be saved?
11. How is death regarded?