Why the poem begin with Telemachus? How does his search in The Odyssey develop his character? What kinds of changes do we see with the Greek kings now they are home from the Trojan War?

To answer your second question of how Telemachus' search for his father helps develop his character, let's look at him at the beginning of the poem and at the end.
Telemachus was just an infant when his father left to fight in Troy. When the story starts, he is still a youth, but approaching the cusp of manhood. The first four books of the Odyssey follow Telemachus's search for news of Odysseus and his maturing into a man. When we first meet Telemachus in Book I, he is still clearly an adolescent. He has not yet developed his own sense of agency or self-confidence. He is still a brooding teenager, living in the midst of unruly suitors, powerless to do anything about it. It is up to Athena (disguised as Mentor) to jumpstart his journey into manhood. As she tells him, "You must not cling to your boyhood any longer / it’s time you were a man" (1.341-2).
Telemachus takes up this challenge, but he still has a long way to go. We can still see his youthful brashness in Book II. When standing before the suitors he gives a decent speech but follows it by throwing the speaker's scepter upon the ground and bursting out in tears. In Book III, after having arrived at King Nestor's island in search of news of his father, he is too nervous and self-deprecating to approach the king without Athena's encouragement. We can see he has a lot of growing up to do if he is someday going to be the head of Ithaca.
As Telemachus ventures farther from home and meets with other Greek heroes such as Menelaus, he refines his speech and approach. By the end, he demonstrates the eloquence and comportment of a grown Greek leader. As Menelaus tells him "Not even an older man could speak and do as well. / Your father’s son you are. Your words have all his wisdom" (4.2278-9). We can see that Telemachus is growing in confidence and ability as his search continues.
By the end of the poem, when Odysseus has returned to Ithaca and is hatching and executing plans to retake his household, we can see that Telemachus has matured greatly. Like his father, Telemachus is able to keep a cool and calm demeanor as they pull off a high stakes gambit. On his journey to look for Odysseus, Telemachus has learned the confidence necessary for adult leaders and is able to use this to his advantage to take calculated risks just like his father.