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.
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.
I think that Shakespeare tries to teach us about the importance of social status at the time. A strong theme throughout the play is of the characters honour and faithfulness. Therefore Shakespeare attempts to sow us how important it is for such characters to withhold their honourable positions.
For example Hero is the greatest teaching in the play where after being accused of unfaithfulness still without questioning she goes back to marry Claudio. Personally I feel that this is not done out of true love but instead it is done to only restore her faithfulness and honour. Simalarly other examples of maintaining a social status are shown when Benedick is forced to 'kill Claudio'. Although he accepted the challenge to prove his love for Beatrice i feel he also accepted the challenge to prove his masculinity as he could never back down from a fight.