Compare and contrast the idea of everlasting life or destiny as portrayed in the Epic of Gilgamesh to concepts you are aware exist still today."
Well, one aspect of everlasting life that we see in common is a belief in the afterlife. However, the images of the afterlife in Gilgamesh may be strange to someone raised in a culture with predominantly Judeo-Christian teachings. The underworld is certainly dark, and it appears that all souls alike are trapped underground, to shuffle about like birds. Because this is such a fragmentary look at the beliefs, it is difficult to form an accurate picture of their vision of the underworld.
However, what we do have is a clear description of Gilgamesh's journey to immortality. He is driven by sorrow for the loss of Enkidu, his best friend and equal. This story resonates quite clearly today, for who has not felt grief at the loss of a friend? & who does not question their ideas about mortality, God, and the afterlife when faced with such a tragedy? I am not personally aware of any specific concepts of immortality, unless we discuss legends, such as the philosopher's stone, the fountain of youth, etc. I don;t know if people still believe in these objects, but they could be a parallel to Gilgamesh's search.
As for destiny, many people believe in a form of predestination today. Some believe in an impartial fate, while others believe a deity has decided our lives' courses before the dawn of time. There seems to be a form of destiny at work in the epic, although that destiny can also be changed, depending on whether a god really likes you or not. They have relationships, they pick favorite mortals to guide (or hold grudges against others and attempt to destroy them), they fight amongst themselves. When Enlil chooses to destroy mankind in the Flood, Ea saves Utnapishtim by telling him to build the boat. When the Flood is at its fiercest, Enlil is safe within his palace, while the other gods are cowered around the gates, soaking and miserable. We could compare this story to the flood in the Bible, looking at God's decision to spare Noah along the same lines as Ea's decision to save Utnapishtim.