What does Epic of Gilgamesh suggest were values that were important to the ancient Sumerians?
The Epic of Gilgamesh is in a certain sense a mythical history of how Uruk came to be civilized and how kingship moved from the brute force of an unrestrained strongman to a society governed by law codes in which rulership carried obligations as well as power.
We see a society in which there is a great degree of gender inequality, with women seen as having domestic and sexual roles and men ruling and engaging in hunting and warfare. Marriage is an important institution, and sleeping with another man's wife is considered wrong. Physical strength and attractiveness are admired.
Society is highly stratified, and people are expected to behave in a manner appropriate to their inherited station. That includes both respect for superiors and just treatment of inferiors.
It is important to respect and obey the gods, who may bring disaster upon the individual or the city if they are angered.