How can I learn to look at a quote from the Epic of Gilgamesh and identify the characteristic or term being used?

Many of the common literary terms appear far more complicated than they are. They are basically intended to describe patterns you can find in words, just as terms such as "circle" or "rectangle" describe physical patterns.
The types of patterns described by literary terms are often divided into "figures of sound" and "figures of thought." "Figures of sound" deal with patterns such as rhyme, regular rhythms, and other repeated sounds such as alliteration (the repetition of consonant sounds as in "Peter Piper Picked a Peck of Pickled Peppers"). As you are reading Gilgamesh in translation rather than in the original Akkadian, it's difficult to get sense of how the original text used specific figures of sound, although you will get some sense of how it sounded by listening to it read aloud in the original Akkadian.
The second types of patterns are "figures of thought," which have to do with grammatical patterns or patterns of ideas. Climax, for example, means arranging ideas in order of increasing importance, i.e. in the form X is good, Y is better, and Z is the best." A very common version of climax is a three part structure, called a "tricolon crescendo" (roughly three-part increase in intensity). A typical example of this in Gilgamesh is found in the Prologue: "He saw the Secret, discovered the Hidden, he brought information of (the time) before the Flood." Here we have three parallel phrases, each more important than the preceding one.
Another common type of figure of thought is comparison, trying to illustrate something by comparing it to something else. There are two main figures of comparison, similes, which use comparative words such as "like" or "as," and metaphors, which do not use explicit comparative terms. For example, the author emphasizes Enkidu's strength by saying: "his strength is as mighty as the meteorite"; since this uses the word "as," it is a simile.