Shakespeare is trying to show in Much Ado About Nothing that appearances—and words—are deceiving. It takes time and an ability to persevere to get to the bottom of who a person is, as people are most often not what they might seem.

We definitely shouldn't rely on another person's word, as Claudio does with Don John, allowing him to frame events so that it looks like Hero is unchaste. Claudio in this case does realize that appearances can be deceiving, but he himself is deceived in how this is so. He renounces the faithful and pure Hero for using her outward beauty to deceive him:

O Hero, what a Hero hadst thou been,
If half thy outward graces had been placed
About thy thoughts and counsels of thy heart!
But fare thee well, most foul, most fair!

But, in fact, Hero is "most fair" inside and out, a pure, stereotypically 'good' woman. Shakespeare's message here—as in Othello—is that men have to get beyond relying on other men for their opinions of women and actually get to know these females for themselves.

Likewise, Beatrice and Benedick put up false appearances and deceive themselves until they start to believe in their harsh and witty wordplay putting each other down. They therefore don't realize how much they are in love. They need to be deceived to recognize the truth beneath the surface of their banter that are meant for each other.

The layers of appearance and reality can be dizzying—and seemingly contradictory—but at the heart of the play Shakespeare wants us to be thinking twice and maybe three times about the reality at the heart of our relationships.

One lesson that develops from Much Ado About Nothing is that intentionally bearing false witness against someone--intentionally lying about someone--leads to unhappiness, desperate measures, and deepened wrongdoing. This lesson teaches the value of being morally upright people who don't attempt to defame other people for reasons of selfish gain or self-aggrandizement.

In Much Ado About Nothing, as in most Shakespeare plays, the star of the show is the language: its beautiful poetry and prose, its use in clever wordplay, its use as a means of wooing, its use as a means of deceiving, its use as a means of destroying and then restoring honor.  So, Shakespeare is teaching us the power of language in this comedy.

He is also teaching us that there problems in the social structure, the treatment of women, and male reputation.  Any time a man like Don John can deceive a Prince and his friends to defame a woman's honor, then there are the problems in the social class power structure.  Shakespeare is revealing how sexist the culture is.  So says Enotes:

Given the evident centrality of the relationship between Beatrice and Benedick to the narrative line of Much Ado, one of the salient themes of the play necessarily revolves around gender, gender roles, and the differences between men and women.

Ironically, it takes a low-ranking member of the social class to expose Don John and his accomplices.  Dogberry and his motley crew, perhaps not as protective of male honor as Don Pedro and Claudio, see through Conrad and Borachio's male pride, exposing them to Leonato.

So, Shakespeare shows us that women are often mistreated by well-meaning but prideful men.  He shows us that most in the society hide behind reputation, using or abusing it to their advantage.  Above all, Shakespeare teaches us to laugh: at the language, at the role reversals, at the verbal sparring, and even at Dogberry's butchering of the Queen's English.