Gilgamesh has watched Enkidu die and refuses to allow the final rituals to remove his body. Instead Gilgamesh mourns Enkidu for a long time watching all the horrible processes of death consume his friend. Afterward, Gilgamesh is so frightened by death and by the prospect of his own death, his own mortality, that he vows to find the way to immortality. It is for this reason that he seeks Utnapishtim who was granted immortality by Enlil. Gilgamesh wants to make Utnaoishtim share the secret of attaining immortality with him. Gilgamesh endures a horrible journey to find Utnapishtim and in the end returns to his kingdom a wise and kind ruler.
The Epic of Gilgamesh follows the course of the hero's journey twice. In the first iteration of the cycle, Gilgamesh meets Enkidu.
Enkidu is more than just kin to Gilgamesh, he is a kindred spirit and functions as an element of conscience for the unruly and proud king. The journey for Gilgamesh in this first iteration of the hero's quest is one that we might characterize as being concerned with achieving adulthood or realizing one's proper place in the social world. Gilgamesh is half-god and so his limitations are not the same as the limitations of his subjects, the citizens of Uruk.
His early bad behavior is an expression of adolescence, to phrase the situation in contemporary terms. Gilgamesh does not know right from wrong and tests his limits incessantly. Enkidu's arrival tempers Gilgamesh and, ironically, the wild man Enkidu serves to socialize the king. The subsequent journey into the woods to slay the monster there functions as a proving ground, as it were, for the newly adult Gilgamesh. He acts rashly in this episode, fighting Humbaba, but ultimately puts away his childhood and has no more need for his outwardly symbolized conscience. He has grown up and returns to Uruk as a person who understands responsibility and loss.
The second iteration of the hero's journey in The Epic of Gilgamesh depicts a quest for wisdom. When Gilgamesh sets out to find Utnapishtim, he believes that he is searching for immortality. Ultimately, this proves impossible. It was impossible all along. Again, Gilgamesh realizes his limitations - he is mortal too - and in doing this attains yet another stage of development, finding the wisdom of age.
Thus, two cycles of the hero's journey take Gilgamesh from a relatively ignorant state of childishness where he does not understand social limitations to a final state of wisdom and age wherein Gilgamesh has realized both his social limits (attaining a sense of morality) and the limits of mortal life (attaining a sense of scope and humility).
Reading the tale this way certainly simplifies the story-line into a rather rigid symbolism. Karen Hardison has provided another compelling interpretation of the story in her post above, which uses a character-psychology model as a basis of interpretation (one that agrees with this kind of "stages of development" reading).
Each of these modes of interpretation take to heart the notion that the story has two levels of meaning.
"...the journeys that structure the Epic of Gilgamesh need to be read on two levels: first, at the narrative level of physical action in the world, and second, at the symbolic level of supernatural meaning and fulfillment" (eNotes).