In order from start to finish, in Homer's The Odyssey, did Odysseus travel starting in Troy and ending up in Ithaca? The book is The Odyssey by Homer, translated by Robert Fitzgerald.
In fact, Odysseus left his home in Ithaca twenty years prior to the beginning of this tale, and when The Odyssey begins, he has been "on the road" for ten years, trying to make his way home again.
This poem is an epic, meaning that it is very long. It is filled with amazing adventures, the gods, and magical creatures that have kept Odysseus from being reunited with his family. Calypso alone, for example, keeps Odysseus for many years before he is able to ultimately leave her. He visits the land of the Lotus Eaters and the island of the Cyclopes. He angers some gods (for example, Poseidon), yet is protected by others, like Athena, Zeus' daughter.
When all is said and done, the end of Odysseus' story takes place in Ithaca, where suitors, believing Odysseus is dead, have descended on his home, vying for Penelope's hand as if she were a widow. They have come uninvited, have eaten her food, and lorded themselves over her and her household; ultimately, Penelope gives them a challenge. The winner, she says, she will marry.
Of course, Homer does not disappoint, as Odysseus finally returns home, disguised as a beggar, to try his hand at the challenge, though the suitors insult him. Odysseus defeats all the other "contestants," and kills all the suitors for their disrespect to his wife and home. After a disbelieving Penelope is convinced this man is her husband, the couple is reunited.
The tale begins with Odysseus' trip to Troy to do battle, and ends with his return home to Ithaca.