Looking at “The Apology” as Socrates’ last chance to teach the Athenians something, what do you think he tried to teach them?

Socrates's main goal was to help the Athenians, specifically the council that ruled over them, realize some irrefutable truths about justice. He had been accused of teaching false religion to the youth, and while it is true that Socrates may have believed in one god, he did not enforce this idea on his students. Rather, he taught them to question things logically for themselves.
At a typical Athenian trial, the accused was expected to make a speech, after which the council would generally reduce their sentence. Socrates refused to admit guilt. He told them, "I would rather die having spoken in my own manner than speak in your manner and live. . . . The difficulty, my friends, is not to avoid death, but to avoid unrighteousness." His lesson here is that we should never be untrue to our ideals just to save our own life. And these men were doing much more than that; they were putting an innocent person to death simply because they were too stubborn and afraid to admit that they were wrong. What guilt this honest point must have laid on some of Socrates's accusers, who were putting to death a seventy-year-old man who had been of great service to Athens. (In fact, Socrates had the cheek to suggest to them that his service to their country actually called for a pension.)
Continuing on the theme of justice, the philosopher-teacher told the jury of 501 citizens, "The easiest and noblest way is not to be disabling others, but to be improving yourselves." In fact, he asks his condemners to help his sons grow to be “just” men:

"Punish them . . . if they seem to care about riches, or anything, more than about virtue; or if they pretend to be something when they are really nothing—reprove them, as I have reproved you, for not caring about that for which they ought to care. . . . And if you do this, both I and my sons will have received justice at your hands."

Socrates is basically saying that if his sons ever become like these accusers, they will be unjust and deserve punishment.
Socrates's words touched the hearts and minds of those citizens who voted him innocent, yet he was found guilty by a vote of 280 to 221. However, his words, recorded in the writings of his student, Plato, immortalized Socrates's ideas and influenced humanity's views on justice for every generation thereafter.

The first thing Socrates is advocating is free speech. He believes that free inquiry is a crucial step in attaining knowledge, and even under the threat of death, he refuses to be silenced or censored.
Equally crucial is the importance of knowledge and truth. To be a good person, one must know what is good and true. This means that we should constantly examine ourselves and other people, searching not just for commonly accepted beliefs but for actual empirical evidence and logical correctness.
Next, Socrates believes in the importance of genuine expertise. One would not entrust captaining a ship to a landlubber who had never been to sea, so important decisions in politics, education, and justice should be in the hands of people who have profound knowledge of those subjects rather than demagogues.
Finally, Socrates believes that ideals are important and that sticking to one's moral principles is more important than even one's life.