It's very likely that the Greeks had puppets--or more accurately marionnettes--and put on shows. Certainly their children had wooden dolls with articulated arms and legs that would have been suitable for the purpose with the additions of only a few strings.
The first known mention comes from Herodotus, where he says the Egyptians had a figure, certain parts of which could be moved by a string. The word he uses for this is neurospasta, which means something like string-puller. Next, Xenophon, in his Symposium, has Socrates present when a troupe of entertainers puts on a show that includes neurospasta. The same word. On the face of it, this is case closed, except by context it might also refer to the sinews of a boy contortionist who is mentioned as being present. The third mention is Plato, who in his Laws refers to children being entertained by a thaumiston, which means something like a show of wonders. Most people take this to mean a puppet show. There are enough other mentions in passing that it seems a safe assumption.
Assuming they did have marionnettes, no one knows what stories they told, except that it can't have been Punch and Judy: that's a much later creation.