A fantastic source for your question is Robin Osborne, Greece in the Making, 1200-479 BC 2nd edition (Routledge 2009). He traces the beginnings of Greek identity in the aftermath of the end of the Bronze Age, discussing along the way many of the cultural aspects which later defined the Hellenes. A very excellent narrative that employs historical as well as archaeological evidence.

In 600, "Greece" is a collection of city-states (poleis), which have been going strong since the 8th century. It is too early for a select few, including Athens, to develop into the Classical democracies. 600 is in the middle of what we call the Archaic period of Greece, when the default city-state government is the oligarchy (rule by few) or the tyranny (rule by one). Social order is developed, typically with an elite ruling class based on wealth and prestige, propped up by social domination and laws, which by 600 are often being inscribed on bronze or stone and displayed in the community. There is a loose consensus among the cities on how Greek should be written and spelled, though dialects cause a lot of local variation. The alphabet has been transmitted via speakers and writers of the semetic Phoenician language, probably in the 9th or 8th century BCE (our oldest surviving example of Greek with this new alphabet dates to about 750 BCE). How exactly the alphabet was transferred is still a mystery.

The polis usually has a city center (an astu) surrounded by a territory under its direct control (a chora). Some are bigger, some are smaller. We have come to accept that the boundaries of the chora were often determined by extra-mural (outside the walls) sanctuaries, and the procession between the center and the sanctuary. Some good examples are Athens -> Eleusis, or Argos -> the Heraion (temple of Hera). Every polis tended to have its own particular local gods, and sanctuaries to those gods. Athens had Athena, of course; Argos had Hera; Sparta had Artemis Orthia. By 600, many of the boundaries of the city-states had been more or less solidified, and the fighting over those borders was in full swing. When not observing the gods at home, the Greek city-states are sending out delegations to athletic competitions at the Pan-Hellenic sanctuaries at Delphi, Nemea, Ischia, and Olympia. By 600, these places have become the site of of friendly rivalry between both city-states and individuals, and it has become accepted practice for the elites of the various cities to display their wealth and to compete with each other by their dedications and sumptuous banquets. Soon, the cities will begin to construct "treasuries" at these sanctuaries to announce their power and wealth to the pan-Hellenic world.

Polis society in the Archaic period is divided between the "have alot," the "have enough," and the "have little." Farmers were firmly in the middle category, and they made up the bulk of the military forces. They carried their own weapons and armor, the distinctive shield and spear and helmet, and we call them "hoplites." Hoplites will be a feature of Classical Greece as well, but it is in the Archaic period that we hear about the most hoplite warfare, typically two city-states fighting it out over some dispute or other. At the bottom of the social hierarchy were slaves, who were distributed throughout Greek society, both in the urban and in agricultural sphere.

In terms of literature, 600 is too early for the great Classical dramatists: Tragedy and Comedy are not around yet, so no Aeschylos or Sophocles or Euripides or Aristophanes. This is also too early for the familiar Athenian philosophers like Socrates, Plato, and Aristotle. But in 600, the Greeks are writing lyric poetry (like Sappho, the poetess) and are very much enjoying the Homeric epics. They are performing Homeric hymns to the gods, and creating new ones; and they are singing songs about love, about society, about warfare.

By 600, the area which is "Greek" has expanded to encompass not only the mainland centers (like Sparta, Argos, Corinth, Athens, and Thebes) but also the West and East. In the West, the Greeks have ventured to found new enclaves in Sicily (Syracuse, e.g.), southern Italy (Tarentum or Metaponto, e.g.), Western Italy (Cumae, e.g.), and southern France (Marseilles). In the East, the Aegean coast of Asia Minor (Turkey) is now rife with flourishing Greek merchant cities like Miletos and Ephesos.

These are just some of the aspects of Greek life at or around 600. For the full picture, please do check out Osborne's book. It's great!