The Greeks did not have the same idea of an economy that we have. The word “economy” is Greek, but to the Greeks it meant something like “rules of a household” (the “eco” part of economy is from the Greek word for house, “oikos”, and the “nomy” part is from their word for law). So the economy was the way a household ran, and also the way a city-state ran.
Even as far back as the Stone Age, many Greeks were sailors, and sailed all over the Eastern Mediterranean. Like many other sailors in other places and times (like the Vikings for example), Greek sailors found a lot of different ways to make their living from sailing. Some of them were fishermen, and ate some fish and sold some in markets. Some were pearl-divers. Other Greeks were traders, who bought things at one port and sold them at another port, and made some profit for themselves along the way. Many Greek people made things for the traders to sell: wool cloth, wine, perfume, and fancy pottery. Other Greeks were soldiers for their city-state, who conquered other cities and forced them to pay tribute. Many Greek sailors worked as mercenaries, hiring out themselves and their ships to fight for other countries like Egypt.
Other Greeks were pirates, who simply raided wherever they could and took whatever they could get. In real life, people probably didn’t fit so neatly into any of these categories. Pirates sometimes traded, and sometimes fished, and sometimes hired themselves out as mercenaries. Traders were not above doing a little raiding if they got the chance. For soldiers, the difference between fighting and raiding is not always very clear.
One reason for raiding was to capture women as prisoners of war. Like other people – like the Assyrians or the Egyptians or the Persians – Greek rulers enslaved thousands of women to work in giant weaving workshops, spinning and weaving expensive wool cloth that traders could carry on their boats and sell.
Trade & Warfare, by Robert Hull (2000).
Ancient Greek Jobs (People in the Past Series-Greece), by Haydn Middleton (2002). Easy reading.
The Ancient Economy by Walter Scheidel, Sitta Von Reden (2002). A collection of essays by different specialists, but written for the non-specialist.
Economy and Society in Ancient Greece, by Moses Finley (revised edition 1983)
Economic and Social History of Ancient Greece: An Introduction by M.M. Austin and P. Vidal-Naquet (1980)
The Ancient Economy, by Moses Finley (1973, revised edition 1999). This has been the starting point for academic discussions of the Greek and Roman economy since it first came out thirty years ago.