What does Athena advise Telemachus to do?
Athena's advice to Telemachus can be found in Book 1, lines 296-97. She is telling him that the time has come to leave his childhood behind and be a man, saying, “…You should not go on clinging to your childhood. You are no longer of an age to do that."
Dr. Walter Englert, of Reed College, explains the advice very well. He says, "These words could be a direct quotation of a mother or father dropping their hesitant son or daughter off at college, and with them Athena is giving Telemachus needed advice. Telemachus is in a tough position as the Odyssey opens, and he needs to grow up. When Athena speaks these words to Telemachus, his father Odysseus has been away for 19 years. He left when Telemachus was just a baby, and Telemachus has no memories of his father. For the last three years, the situation in their house has been desperate: 108 suitors, convinced that his father Odysseus will not return and attracted by the beauty and intelligence of his mother Penelope, have been living at the palace in a riotous three-year-long party. Violating all the accepted rules of the guest-house relationship and ancient courting practices, they have been consuming his household’s food and drink and sleeping with the female servants, trying to force Penelope to give up waiting for Odysseus and marry one of them. Athena is right. Telemachus needs to grow up quickly."
In books 1-3 Athena acts as a mentor to Telemachus just as she has done for Odysseuess so often in his journeys. Athena guides Telemachus when he calls together the council to address the insult that the suitors have made upon him and his mother. Athena arranges this to show Telemachus humility and that the decisions made by a king are not always popular among the people. Here, Athena teaches Telemachus that it is time to start being a man. Her major concern for Telemachus is his safety and making sure he is properly following the footsteps of his father. By sending him on his voyage, Athena ensures that the suitors cannot kill him and this is justified later when she warns him that the suitors are ready to ambush him on his return. Furthermore, by Telemachus going to Pylos and Sparta he is beginning diplomacy with the countries visiting his father’s old war companions, and setting up possible allies in case Telemachus requires support for the crown.
Note the two answers above. In addition, though, note that Athena could, supposedly, tell Telemachus that his father is safe, and protect Telemachus with her godly powers. But that's aside from the point. Telemachus has to achieve his manhood by himself. While she does advise and guide him (disguised as Mentor, his tutor), the adventure itself must needs be of his own doing.
Note, too, that the Odyssey is the story of a homecoming. Telemachus' journey to Pylos and Sparta parallels his father's journey. Each - father and son - must leave home and go on a quest before returning victorious.