It's clear that the story of Gilgamesh—the historical king of Uruk—was handed down from one generation to the next, sort of like a game of telephone, emphasizing different characteristics and storylines over time. (We are fairly certain that the original King Gilgamesh wasn't actually 2/3rds god and didn't really travel to the edge of the earth.)
As we discuss in the "Writing Style" section of this module, The Epic of Gilgamesh went through many different incarnations before reaching its present version. In fact, it seems to have been a dominant story for over 1000 years throughout the region of Mesopotamia; therefore, we think the most basic classification for this story is in the "Folklore, Legend, and Mythology" category. (In fact, we think it might have invented this genre.)
But we also think we can stick it under "Epic Poetry"—and, we have even better reasons for saying so than simply that it is called The EPIC of Gilgamesh (though, we agree that's a pretty good reason). Like, epic heroes:
Obviously, Gilgamesh has a pretty good claim to be an "Adventure," because of all the crazy derring-do that transpires. Plus, it's definitely a "Quest"—the whole second half of the story is devoted to Gilgamesh's quest for immortality and the above-mentioned derring-do that ensues.
And finally? We think The Epic of Gilgamesh is also a "Coming-of-Age" story because of the way it shows (1) Enkidu leaving the wilderness and coming to know the ways of human beings, and (2) Gilgamesh developing from a pampered egomaniac into somebody who knows the value of true friendship, and who understands the importance of life and the necessity of death.