In How to Read Literature Like a Professor, Thomas C. Foster tells us that in order to have a good quest you need a few ingredients: a quester, a place to go, a reason to go, an obstacle, and a real reason to go (because the stated reason you are going is never the real reason).
Almost every action in this poem deals with some kind of journey or quest:
Enkidu journeys from the wilderness to Uruk, making a stop off with Shamhat, of course.
Gilgamesh and Enkidu go on a quest to the Cedar Forest—with the stated reason that they shall seek fame. What do you think is the real reason they go? Is it to solidify their friendship? To prove their faithfulness to each other? To demonstrate their power over nature, now that Enkidu is a semi-civilized critter?
Then, there is Enkidu's quest to the underworld—through the twin-peaked mountain Mashu, to Siduri's tavern, across the sea of death with Urshanabi—to find Utanapishtim and discover the secret to immortality. But, that isn't the reason that he goes. Of course, that is why he thinks he goes, but really he has to go on that quest so that he can discover that, while no man can have immortality, a selfless, devoted king can become immortal to his people (and literature readers everywhere) simply by being a good leader.