Final answer:
Telemachus sets out on his journey encouraged by Athena, both to search for his long-lost father, Odysseus, and to progress into adulthood. This quest holds the promise of restoring honor to his household and allowing him to establish his own heroic identity.
Step-by-step explanation:
Telemachus, the son of Penelope and Odysseus, embarks on his journey largely due to the influence of the goddess Athena, who appears to him as Mentor. She persuades Telemachus that he must search for his father, who is believed to be still alive after the fall of Troy. This decision is not made lightly, as it involves leaving his mother Penelope to deal with the difficult situation at home, where assertive suitors are trying to pressure her into marriage, believing Odysseus to be dead. However, Athena's guidance implies a greater good that could come from finding Odysseus — not only might he be able to return and save their home but it will also allow Telemachus to grow and establish himself independently.
Telemachus sets sail to learn about his father’s fate, driven by a sense of duty and the hope of dispelling the dishonor brought upon his house. Importantly, Telemachus is faced with the transition from youth to adulthood, and this journey signifies a rite of passage — an important step towards asserting his own place in the world and potentially becoming a hero in his own right.