Actually, if you research Euripides you will know that he was not well accepted among his peers or Greek society in general, as he tended to speak up for the voiceless in his society (namely Women and "Barbarians"). Remember that this was a patriarchal society, and taking up the cause of women in his plays was not the popular thing to do.
In fact, Euripides's plays were not popular until after his death when his son introduced them into a Dionysus play festival where they finally caught on.
A Greek audience of this play would have already known the outcome -- it is based on the myth. As Euripedes writes the story, he uses the character of the Chorus to be the voice of the people. In reviewing the Chorus interludes we can see the change in their attitude toward Medea. At first they are just concerned over her sadness and anger. After listening to Medea, they are much more sympathic and supportive or her drive for revenge. They support her until they realize she intends to kill her children. Then the Chorus tries to talk her out of killing her children, asking how she could possibly live with the guilt. The Chorus is a sympathic voice of reason, and would feel a lot like the Greek and modern audiences who see the play.
Even though we hear stories about it more and more often, the shock of what Medea did would have been the same in the time the play was written as it is for us today. It is unthinkable that a mother would murder her own children. In most of the cases we hear about today, such as Andrea Yates, the mother suffers from some form of mental illness. Medea, however, suffered from the proverbial green-eyed monster of jealousy. Her husband, Jason, had abandoned her for another woman. To get back at him, she kills "his" children. In the play, Jason calls Medea a "most utterly hateful woman" and says that "no Greek woman would dare do this." That says it all.