As other contributors have already expressed, The Epic of Gilgamesh celebrates friendship (via the friendship between Gilgamesh and Enkidu), as well as personal bravery and strength (via Gilgamesh's various triumphs). Based on this ancient story, you might expect ancient Sumerian culture to have a certain aristocratic mentality, which celebrates personal...
excellence and achievement.
In addition, however, I get the impression that The Epic of Gilgamesh also contains a deeply pessimistic vision on the human condition. Indeed, it's worth noting that Gilgamesh's great quest to achieve immortality ends in failure, and while he is able to attain a plant that holds restorative powers (which would reverse the effects of aging), he loses that plant when a snake eats it. For all of his many successes, he ultimately fails in his greatest and most ambitious endeavor. Thus, the story makes a clear statement: human lives are transient, and it is impossible for any human being to overcome their mortal nature.
Gilgamesh has to learn this truth (and ultimately accept it). For all his strength, death is something he cannot overcome and cannot fight. From that perspective, the poem seems to suggest a somewhat complicated vision of excellence, which, even as it encourages people to strive for greatness, demands that they simultaneously recognize their own limitations as human beings.