Discuss several ways in which Homer uses his tale to illustrate moral values that were important to the culture of his time.

The Odyssey is a major Homeric work, and it certainly displays the moral values of Homer's time.
In the story, Odysseus is the quintessential Homeric hero. He is loyal, courageous, honorable, tenacious, and principled. These traits are also present in other characters such as Telemachus, Penelope, Philoetius, and Eumaeus.
Penelope remains faithful to Odysseus, despite the fact that twenty years have passed. She refuses to marry any of the suitors that have taken up residence in their family home. Telemachus also remains faithful to his father, despite the fact that suitors like Eurymachus have derided him for his loyalty. In fact, Telemachus tells the suitors that he will put up with them wasting his estate for another twelve months, provided that he can get confirmation his father still lives. Telemachus is willing to brave every trial and endure every privation to discover Odysseus's whereabouts.
Loyalty is emphasized in the story because it also reinforces an important cultural value in Homer's time: the integrity of the family structure and its significance to ordered society. Penelope, despite her helplessness, holds her ground against the suitors. She may not challenge any of the men physically, but she demonstrates stoic resolve by putting them at an emotional distance. She uses verbal and indirect means to preserve her virtue and is rewarded for her steadfast courage (both by the gods and her husband).
Penelope and Telemachus are not the only ones who remain loyal to Odysseus, of course. The swineherd, Eumaeus, is the first person Odysseus sees when he returns (although the faithful servant does not initially recognize his master). Despite the suitors' threats against Telemachus, Eumaeus stands his ground, giving unequivocal support to Telemachus and Penelope during Odysseus's absence. He is as principled as Odysseus and Telemachus.
An important cultural value in Homer's time centers on a reverence for the major Greek gods. In the story, faithful and principled characters are shown to rely steadfastly on the mercies of gods like Zeus, Hermes, and Calypso. The text tells us that Eumaeus sacrifices his best pig to make a meal for Odysseus (while still a stranger to his servant). Before they eat, Eumaeus gives to every god his portion. Thus, Eumaeus is shown to be equitable and honorable, a "man of good principles."
The reverence for the gods in The Odyssey demonstrates the moral compass by which ancient Greeks ordered their lives. The story even begins with a convocation of gods. Homer shows that not only are the Greeks mindful of their gods but the deities are also highly interested in human affairs on earth. It is Athena who beseeches Zeus to aid Odysseus in his quest to return home.
The gods recognize personal courage, generosity, and fidelity as worthy traits. As such, they are willing to provide every advantage to those who exhibit such excellencies in their character. So, the above are some of the ways Homer uses his story to illustrate the cultural and moral values of his time.