Which quotes from the Odyssey demonstrate that Odysseus is using what he has learned?
From my perspective, the key moment in the Odyssey that shows the greatest effect on Odysseus later on is his revelation of his identity to the Cyclops at the end of Book 9. Thus, I would say that if we want to find quotations that indicate that Odysseus is using what he learned, then we should look at places where Odysseus conceals his identity.
We can find examples of this in Odyssey 13-16, where Odysseus conceals his identity from his faithful swineherd Eumaeus. We can also see this in the fact that Odysseus is disguised as a beggar when he first returns to his own home and has to endure the abuse of another beggar, as well as the suitors. Odysseus does not reveal himself to the suitors until the perfect moment. Odysseus even conceals his identity from his own wife and father until just the right time.
So, I would encourage students to look at these passages for a quotations like the one that follows, where the disguised Odysseus tells a story to Eumaeus about Odysseus:
But in the third watch of night, when the stars had passed the zenith, I nudged Odysseus who lay next to me, with my elbow, and he listened while I spoke: ‘Odysseus of many resources, scion of Zeus, son of Laertes, I am not long for this life: I’ve no cloak, and the cold is killing me. Some god persuaded me to wear a tunic only, and there’s no escaping this frost.’