Badass Boast: After Gilgamesh and Enkidu destroy the Bull of Heaven - the divine beast sent to destroy them for mocking the goddess Ishtar - Enkidu shakes the bull's torn off haunches at Ishtar, threatening to do the same to her if he catches her.
Designated Hero: Gilgamesh is a tyrant whose outrages against his people drive them to pray to the gods for aid. Later, he and Enkidu kill Humbaba solely to win glory, then anger the gods with their arrogant actions. By 21st century standards, Gilgamesh is not very heroic.
Memetic Badass: The sole reason why the story survives today. We wouldn't have so many tablets if they weren't being made and put all over Mesopotamia just to tell everyone how badass he was and how you can never hope to reach the level of sheer badassery he was at.
Fridge Brilliance can be applied here when you figure that well, though he may have failed to achieve literal immortality within his own story; the Epic of Gilgamesh in the meta sense has become immortal and survives to this day.
Despite all of his previous outrages, Gilgamesh is accepted back as king of Uruk at the end of the epic, with neither humans nor gods nursing resentment against him. Many 21st century readers would find this unbelievable.
The tendency to paint Shamhat as a harlot and seductress. In the story, her sexual encounter with Enkidu humanizes him. Her week-long tryst with Enkidu isn't a fall, but the first event that raises him above the level of an animal.
Despite Ishtar's less-than-flattering portrayal in this story, she was a very important and revered goddess and sex is part of the sacred duties of her priestesses.
What an Idiot!: If you have a plant that will make you immortal, don't wait until after you've taken a bath to use it.
He was actually going to give it to an old man to see if it would turn him young again, so he was probably playing it safe since he didn't know if it would actually kill him instead.
Woolseyism: Since some portions of the story have been lost, some translations feature original material to fill the gaps which occasionally works quite well. Most notably, at least one changes Gilgamesh's motivation from wanting immortality for himself to wanting to bring Enkidu back to life.