Before Alexander the Great, each city was as you mentioned above a city state or an autonomous mini-country with its own language - that is the dialect of Greek; Attic for Athens, Doric for Sparta, Ionic for Eastern Greek Cities like Smyrna, Ephesus, Miletus and lastly Aeolian for the Island of Lesbos- Where the female poet Sappho is from). They also had their own citizen body (demos). So the city included then a place for public speaking and tragedies (dramatic shows) and comedies called the theater, an agora or market place (where people met and talked as well, bought things, discussed philosophy), a gymnasium, and many other buildings surrounding these areas. Outside the city was the cemetery or necropolis (city of the dead). Every city was different but these are the general things that were included - if this is what you meant that is.
