The first four books of The Odyssey are known as the Telemachiad. It is called thusly because these books are concerned mainly with the plight of Odysseus' grown son, Telemachus. Odysseus only appears as a character talked about, rather than a main actor (Odysseus shows up later, in Book V).
Telemachus, who is now twenty years old and does not remember his father (as Odysseus left for war when he was only a babe in arms) is understandably worried for his father. But this is more than simple family anxiety; Penelope, Telemachus' mother and Odysseus' wife, is being pressured by suitors to remarry. This would not only sully his father's memory (in Penelope's and Telemachus' hearts, for they are both loyal to him though they have heard nothing from him for many years), but it presents an economic and succession problem as well. If Penelope remarried Telemachus, the only son of Odysseus the King of Ithaca, would be, in effect, disinherited. Telemachus, now a man, is in such a state of worry and anxiety that he does everything he can to find out what has happened to his father, including what happened in the Trojan War.
In Book I we find out that Odysseus is on the goddess Calypso's island. He has been there for part of the time of the ten years since the end of the Trojan War (which was ten years in length itself, thus making up the the twenty years of Telemachus' life.) Talk of the Trojan War itself does not occur until Book III, in which old King Nestor relates the end of the war and the return home of Menelaus and Agamemnon, but he knows nothing of what happened to Odyseus after the departure from Troy after the sacking of the city.
Telemachus learns a little more in Sparta, where he goes to talk to Menelaus and Helen. Menelaus tells Telemachus that he did learn from the Egyptians that Odysseus was alive some years after the war's end. Helen also retells some of Odysseus' exploits before the walls of Troy:
He covered himself with wounds and bruises, dressed himself all in rags, and entered the enemy's city looking like a menial or a beggar, and quite different from what he did when he was among his own people. In this disguise he entered the city of Troy, and no one said anything to him. I alone recognized him and began to question him, but he was too cunning for me. When, however, I had washed and anointed him and had given him clothes, and after I had sworn a solemn oath not to betray him to the Trojans till he had got safely back to his own camp and to the ships, he told me all that the Achaeans meant to do. (Book IV)
Menelaus remembers, too:
What endurance too, and what courage he displayed within the wooden horse, wherein all the bravest of the Argives were lying in wait to bring death and destruction upon the Trojans. ... Three times did you go all round our hiding place and pat it; you called our chiefs each by his own name, and mimicked all our wives—Diomed, Odysseus, and I from our seats inside heard what a noise you made. Diomed and I could not make up our minds whether to spring out then and there, or to answer you from inside, but Odysseus held us all in check, so we sat quite still, all except Anticlus, who was beginning to answer you, when Odysseus clapped his two brawny hands over his mouth, and kept them there. (Book IV)
Telemachus learns some things about the war (the fall of Troy, the Trojan Horse, the departure for home), but mostly he learns that Odysseus is probably still alive.