Euripides' Medea makes several points about marriage and loyalty. Relatively early in the play, Medea says:
Of all creatures that can feel and think,
we women are the worst treated things alive. (31)
Although the Athenian audience did not believe in equality for men and women, there is certainly a pragmatic lesson that an unhappy wife may wreak vengeance against the husband who mistreats her. Some members of the audience may have understood this as a warning that women should not be allowed power; others may have read it as advocating better treatment of wives.
Another important lesson is that injustice (Medea's sacrifice of her brother, Jason's jilting Medea) leads to more injustice; vengeance is never final but merely perpetuates a cycle of violence.
The method by which drama teaches is embodiment of moral precepts in character and circumstance, a form of argument by example.
I believe the play should teach us to recognize the reality that some women can be so filled with resentment and hatred for a man who has abandoned or otherwise mistreated them that they will turn their anger on the children he fathered. If the children are boys, the mother can merely reject them or go so far as to abuse them mentally or physically, or both. Some mothers may disguise verbal or physical abuse as discipline or instruction. Young children are easy victims because they trust their mothers and want their mothers' affection. They are likely to believe that they are responsible for the rejection or abuse they receive; they can develop lifelong guilt or inferiority complexes as a result. If the children are girls, the mother may condition them in various subtle or overt ways to feel a hatred of men which could make the girls' lives as unhappy as their mother's. Miss Havisham in Charles Dickens' novel Great Expectations is an example of how a mother-figure can teach a girl to share her hatred and desire for revenge against the male race. Another good example is the mother in D. H. Lawrence's short story "The Rocking-Horse Winner."
There was a woman who was beautiful, who started with all the advantages, yet she had no luck. She married for love, and the love turned to dust. She had bonny children, yet she felt they had been thrust upon her, and she could not love them. They looked at her coldly, as if they were finding fault with her. And hurriedly she felt she must cover up some fault in herself. Yet what it was that she must cover up she never knew. Nevertheless, when her children were present, she always felt the centre of her heart go hard.
A really harrowing story of abuse of a child by a mother is the memoir A Child Called "It." (See the reference link to the e-Notes summary of this book below.) Extreme hatred can lead to insanity, and insanity can lead to anything, including murder. Medea is not just a story about one woman but an extreme example of a fairly common type of human behavior, much of it hidden behind closed doors.