Telemachus "left his spirit filled with nerve and courage, charged with his father’s memory more than ever now." In Book II, Athena (disguised as Mentor) becomes like a mentor. She inspires Telemachus again and gives him hope regarding his upcoming voyage. She even acquires the ship and resources he will need. 

In Book III, when Telemachus is feeling despondent and skeptical at the prospects of finding his father alive, Athena reveals that she fully supports Telemachus in his quest. When she does this, Nestor also shows his support: 

“Dear boy—never fear you’ll be a coward or defenseless, not if at your young age the gods will guard you so." 

A good bit of this "training" is giving Telemachus support and a sense of hope. 

In book 2, Athena helps Telemachus in several ways. First, when Telemachus addresses the people, he approaches them as a young prince with all of the immaturity and inexperience he has. However, Athena graces him with power to control his speech and impress the people.

Although the people are impressed, some dissenters make this difficult. Later in the book, she helps again by appearing as a local advisor, Mentor. Telemachus listens to Mentor (who is really Athena) and grows excited to complete a journey or search in the effort to find Odysseus.

A little later, she disguises herself as Telemachus and prepares the entire ship for the journey.

Therefore, Athena motivates, encourages, and serves Telemachus making his quest possible. Each of her three acts contributed to his ability to take the trip, but he has no idea that she did them. Oh, she also went along disguised as Mentor... that makes a total of four actions to help him.

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.

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.

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."

Some context is in order. The epic The Odyssey starts in medias res. The Trojan war has ended and Odysseus is trapped on an island as Calypso's love interest. During this time, Odysseus's estate is being taken advantage of by suitors, who are also seeking to woo his wife Penelope. Many are also saying that Odysseus is dead; all the Greeks are back except for Odysseus. 

Within this context, Telemachus, the son of Odysseus, almost gives up hope, until Athena (with the permission of Zeus) comes down in the disguise of his father's friend, Mentes, to offer counsel. He tells Telemachus to travel to Pylos and Sparta.

At Pylos, Telemachus meets Nestor. This makes sense, because Nestor fought side by side with Odysseus. Moreover, Nestor was known for his wisdom. Unfortunately, Nestor has no information about Odysseus, but he wishes Telemachus good fortune. 

At Sparta, Telemachus meets Menelaus and Helen. They recount many tales of the war and this is where we also hear about the artifice of the "Trojan Horse." They also say that his father is still alive. He has been captured by Calypso on an island.  This bit of information gives Telemachus hope. 

Athena has always been a partisan of the Greeks in the Trojan War (described mostly in the Iliad), and especially Odysseus.  Therefore she is interested in Odysseus' son Telemachus, and she does everything she can to help the young man find his father.  Athena, the patron goddess of, among other things, wisdom, is thought to be partial to Odysseus because of his wiliness and craftiness.  Also, Odysseus, though mortal, was thought to be a hero particularly beloved of the gods.  All this filters down to his son Telemachus, who was certainly in dire straits and in need of the help of a powerful goddess.

Athena was also the goddess of civilized things like handicrafts (such as weaving, which figures so prominently in regard to Telemachus' mother, Penelope).  In a way, Athena can be regarded as a force for home, towns, government, and civilization in general, whereas Odysseus' major godly foe, Poseidon the god of the sea, can be seen as a force of elemental chaos, or perhaps of male wanderlust.  While Poseidon works hard to keep Odysseus away from home, Athena is focused on preserving Odysseus' home (and family, such as Telemachus in this instance) and bringing him back to it.  To put a psychological spin on thing, these gods could be seen to represent the two warring natures within each man; the force which desires home, hearth, and family, and the other wilder side which represents wandering, violence, and freedom.  Athena, who is significantly female, and was one of the greatest of all the gods revered in Odysseus' part of Greece (Athens is named for her) definitely represents the civilized side of life which, after the long Trojan War, Odysseus is longing to return to.

Athena (also spelled Athene) makes Telemachus brave and confident to go to Pylos, though the people and the suitors are against him.

As he thus prayed, Athena came close up to him in the likeness and with the voice of Mentor. “Telemachus,” said she, “if you are made of the same stuff as your father you will be neither fool nor coward henceforward, for Odysseus never broke his word nor left his work half done. If, then, you take after him, your voyage will not be fruitless, but unless you have the blood of Odysseus and of Penelope in your veins I see no likelihood of your succeeding. Sons are seldom as good men as their fathers; they are generally worse, not better; still, as you are not going to be either fool or coward henceforward, and are not entirely without some share of your father's wise discernment, I look with hope upon your undertaking. But mind you never make common cause with any of those foolish suitors, for they have neither sense nor virtue, and give no thought to death and to the doom that will shortly fall on one and all of them, so that they shall perish on the same day. As for your voyage, it shall not be long delayed; your father was such an old friend of mine that I will find you a ship, and will come with you myself. (Odyssey Book II)

Athena, disguised as Mentor, obtains the ship and crew for Telemachus, gives him confidence and the glamor of beauty and strength, throws the suitors into slumber so that Telemachus can escape, and accompanies him on his voyage.  Direct divine help, indeed, and much needed by the young Telemachus.

This question is in reference to the events of book 15. It opens late at night, in Sparta, as Telemachus is struggling to fall asleep (at least, this is the suggestion in the Fagles translation). In this situation, Athena stands over him, ready to spur him to depart. She says that in traveling so far from home, he has left his holdings vulnerable, at the hands of those who would "carve up all your wealth, devour it all, / and then your journey here will come to nothing" (The Odyssey, trans. Robert Fagles, Penguin Classics (paperback ed.): New York, 1996, 319). Therefore, she insists that he get Menelaus to let him leave immediately, as Penelope is already being pressured to wed Eurymachus, whose gifts exceed those of any other Suitor.

At the same time, she warns him about the ambush the Suitors have planned against him and gives him instructions for avoiding it. With that, she departs for Olympus. The next morning, Telemachus will again meet with his hosts and, later still, set off on his journey back to Ithaca.