Telemachus learns a great deal in Book IV of The Odyssey. He arrives in Sparta,with Peisistratos, at the court of King Menelaos and Queen Helen. They were both major characters in the Trojan War drama, and they had both known Odysseus personally. Menelaos and Helen tell Telemachus some war stories about Odysseus, and about what happened to him after the Trojan War. On his return journey home, Menelaos had plotted to trap an immortal, Proteus, for it is said that
If somehow you were able to wait in an ambush and seize [Proteus],
he would inform you about your road and the length of your journey--
and of your homecoming, how on the fish-thronged sea you will travel.
Then too he would inform you, beloved of Zeus, if you wish it,
what has been done in your palace, the evil as well as the good things,
while you have thus been making a lengthy and arduous journey. (IV.387-393)
So Menelaos tells Telemachus of the many things he learned from Proteus while he held him, and among the things he is told is that Odysseus is alive.
So I spoke, and at once he addressed me, giving an answer:
He is the son of Laertes, in Ithaka keeping his dwelling.
I saw him on an island; he shed great tears in abundance
there in the halls of Kalypso, a nymph who keeps him beside her
forcibly. He is unable to leave for the land of his fathers,
since niether well-oared ships does he have near at hand nor companions
who might serve to convey him across the road ack of the deep sea. (IV. 554-60)
So from Menelaos Telemachos learns not only that Odysseus is alive, but also the person who holds him (Kalypso,) and that he is forcibly prevented from returning home. Understandably, this comforts Telemachos, and makes him eager to return to Ithaka.
Iin book three Telemachus learns that Odysseus survived the war and set sail from Troy to go back home. However, Nestor does not have information of Odysseus beyond that; he does offer Telemachus hope that his father may arrive home yet.