What are some of Odysseus's beliefs?

Odysseus is very much a product of his time and place. He is a great king and noble warrior who unhesitatingly subscribes to the dominant ethos of aristocratic Greek males. Yet he does so with a much greater degree of conviction than mere ordinary mortals. For one thing, he is incredibly loyal: to his wife, Penelope; to his kingdom; to his honor; and to the gods.
Loyalty is very important indeed to Odysseus, and he expects the same in return. When it isn't forthcoming, as for example with the suitors and the treacherous maidservants, then revenge is swift and bloody. Odysseus's slaughter of the suitors and Telemachus's hanging of the maidservants represent a defense of the prevailing moral code and the overriding importance of loyalty to one's king and master in ancient Greek society.
Self-control is another of Odysseus's guiding principles. When the beautiful Sirens try to tempt him off course, he orders his men to tie him firmly to the ship's mast so that he won't fall prey to their sweet, bewitching song. When he finally returns home to Ithaca, he takes on the disguise of a beggar and endures kicks and insults from the unfortunate Melanthius and the suitors. But Odysseus bides his time. Revenge is a dish best served cold, and his will have ice crystals all over it.
Talking of which, there's little doubt that Odysseus has a passionate belief in the power of vengeance. Some of us may be a little shocked by the sheer carnage that Odysseus unleashes upon those who've crossed him. But there is method to his apparent madness. Odysseus is a king, don't forget; his is the sole power and authority on Ithaca. So when, for example, the suitors take over his palace, pay court to his wife and later insult Odysseus, they are effectively attacking his throne. This isn't personal for Odysseus; he isn't like Achilles in this regard. His vengeance, though terrible, is totally in keeping with what any aggrieved Greek monarch of the time would demand.
And Ithaca is Odysseus's home. This is where his heart lies. At no point during his epic wanderings does Odysseus ever once consider not returning home to his kingdom. A sense of place and belonging was important for the ancient Greeks, perhaps more so than for us today. Despite their ceaseless journeys and sea voyages, Greeks were firmly rooted to the land of their ancestors. There was no sense of rampant individualism; one's identity was inextricably linked to the role one performed in society. Odysseus's role, as we have seen, is that of king. By returning to Ithaca, his beloved homeland, he has not just reaffirmed his kingship, but also recovered his true identity.