Where does Odysseus' story begin?
The Odyssey begins in medias res, "in the middle of the action." Homer begins near the end and reveals much of Odysseus' adventures via flashback in order to compress time and build toward the climax of the suitors' fates on Ithaca.
In the story proper, we first meet Odysseus on the island of Calypso, his last stop before he washes ashore on the Island of Phaecia, where he narrates his odyssey to King Alcinous.
Depending on how you number Odysseus' stops on his journey, The Odyssey begins at about number 12 (Calypso) and then moves forward to number 13 (Phaecia). Then, it flashes back to the beginning, number 1 (just after the Trojan War). It moves from numbers 2 through 11 before resuming again back at number 13. Then, Homer finishes Odysseus vengeance at Ithaca, stop number 14.
This is a very broad question, because Odysseus has gone through so much that there are many starting points to which we can point. For example, we can speak of Odysseus at the Trojan war and how he was pivotal in the Greek army and how he helped them to win the victory against the Trojansthrough the ploy of the Trojan horse. We read of this in the Iliad of Homer. Or we can speak of Odysseus in his voyage homeward and his struggles along the way. We read of this in the Odessey (again written by Homer). If you want an exhaustive view of Odysseus, read both works.