The temple is the gigantic building at the heart of every ancient city:

Shamhat explains that the temple is where all the people gather at every new year. They take a week off their work, and are taught how and why the city, and hence civilisation, exists, and what the city has planned for the next year. If Enkidu learned this stuff, he might realise why more and more people live in cities, and fewer and fewer people live in the countryside.

Enkidu supports the gods, right? Uruk's temple is owned by Anu (sky god) and Ishtar (goddess of love and war). If Enkidu knew this, maybe he would be curious and could inside. Then maybe he could work out some deal with Gilgamesh instead of trying to kill him?

Should I ask Shamhat exactly what is taught at the new year festival? Should I ask Shamhat who she is? Or just watch and see Shamhat work her magic with Enkidu? Or do something else?