What exactly does Nestor tell Telemachus about the war and the return home?
In the third book of Homer's Odyssey, Telemachus travels to Pylos to visit Nestor, who fought alongside Odysseus at Troy. Nestor tells Telemachus that the Greeks sacked Troy and then prepared to sail away. Unfortunately, a quarrel arose between Agamemnon and Menelaus and the Greeks were divided about whether to sail away immediately (Menelaus) or delay a little while and offer sacrifice to Athene (Agamemnon).
Those who favored Menelaus' view sailed away. Nestor and Odysseus were among this group, but after they sailed a second quarrel led Odysseus to reverse his course and sail back toward Troy. After that point, Nestor did not know what happened to Odysseus. Nestor and Diomedes made it back to their respective homes.
Nestor does relate the rumors that he has heard about other Greeks who made it back home (e.g., Philoctetes, Idomeneus). Nestor also recalls the news that Agamemnon returned home only to be killed by Aegisthus, who, in turn, was killed by Agamemnon's son Orestes. The death of Agamemnon and Orestes' revenge is mentioned throughout the Odyssey and is held up as a parallel to what could possibly happen to Odysseus upon his return home. Odysseus' wife Penelope, however, is faithful, whereas Agamemnon's wife Clytemnestra was the lover of Aegisthus and had participated in the plot against Agamemnon. Orestes, the avenging son, is held up as a model for Telemachus, who ultimately will stand at this father's side as they battle the suitors.
Yet he [Aegisthus] too paid a terrible price, showing how good it is that a man leaves a son behind to take vengeance on his murderer. For Orestes took vengeance on his father’s killer, crafty Aegisthus, for all his treachery. You too, friend, a fine tall man as I see, take courage, so that many a man of generations yet to be born shall praise you. (A.S. Kline translation)