What did the myth of Echo and Narcissus try to teach Greek/Roman society about how they should or shouldn't act?

The myth of Echo and Narcissus has several moral themes, but the main message of the myth warns Greek (and Roman) society about the dangers of pride, vanity, and self-centeredness. In the myth, both Echo and Narcissus suffer downfalls relating to these faults.
Echo is a beautiful nymph, beloved by both Diana and Venus. She has a critical fault however: she talks too much and must have the last word. One day, she uses her talkative nature to deceive and delay the goddess Juno, who is looking for her husband. As punishment, Juno declares that Echo "shall forfeit the use of that tongue with which you have cheated me, except for that one purpose you are so fond of—reply. You shall still have the last word, but no power to speak first." In other words, she can only repeat what she has heard others say. She cannot speak. Echo's personality fault can be seen as a form of vanity. She loves to hear herself speak, and she is punished for it.
One day in the forest, Echo sees Narcissus and falls in love with him. Because of her curse, however, she cannot speak to him directly. She can only repeat back to him fragments of what he says. Narcissus proves to be a vain man, telling Echo, "Hands off! I would rather die than you should have me!" To which Echo can only reply, "Have me!" Narcissus leaves her alone, pining away in love sickness for Narcissus. Eventually she wearies away in her grief, until she is only a disembodied voice in the forest.
Narcissus, like Echo, is cursed in a manner fitting his vanity. Juno causes him to fall in love with himself, and only himself. While gazing into a pool of water, Narcissus is so enamored with himself that he cannot look away. He dies there pining away for himself near the pool of water. Echo, only an invisible voice, stays near him, repeating his lamentations.
So both characters have fatal character faults that lead to their downfall. As in many Greek myths, they are punished by the gods in poetic and fitting ways that match their flaws. These myths would have been an evocative and poignant illustration for Greek and Roman citizens about perils of self-centeredness and vanity. We take the modern word "Narcissism" (meaning "self-obsession") from the male character in this myth.

The myths of Echo and Narcissus are wonderful and do have a moral lesson. Let me first give a short summary.
Echo was a beautiful nymph that loved to talk. One day, Hera was looking for her husband, and Echo believed that he was flirting with the nymphs. So, she held Hera up with her chatter to protect her friends. Hamilton writes:

However, [Hera] was immediately diverted from her investigation by Echo's gay chatter.

When Hera realized what Echo was doing, she cursed her. Echo could have the last word but no longer had the power to speak first. The big problem was that Echo loved the youth Narcissus, but Narcissus scorned all the nymphs, including Echo. In great sadness, Echo pined away to nothing but a voice. As for Narcissus, the nymphs prayed that he would fall in love with one who does not love him. This actually did happen when he fell in love with his own reflection in a pool of water. He, too, pined away and died.
The lessons are twofold. The first is to fear the gods and not cross them as Echo did. It is important to keep in mind that the Romans believed that they were the most pious people on earth. Second, do not act cruelly towards others lest that cruelty turn back on you.