The Ancient Greeks separated politics and religion better than either Mesopotamia or Egypt. Both aspects of their culture played huge roles in their daily lives, however.
In some places, politics were much move involving than in others. For example, the people in Sparta had little to do with the oligarchic government of the city, but the people of democratic Athens all took part in the government (that is, except the women.) In Athens, politics were a very important part of daily life. All men were able to, and expected to, come to the assemblies and help make decisions. They took pride in being able to help change things in their government, and loved to have political discussions. Every man could be leader of Athens for a short time, and every man could be on the Council of 500 (there was a lottery to choose,) which proposed laws and advised the assembly.
Religion was closely tied into everyday life in Greece. There was no strong priesthood in Greece, however. Being a priest was one of many of the other civic duties that citizens took part in. The Greeks, like the Egyptians and Mesopotamians, were polytheistic. Their gods took great interest in the lives of humans, and many myths were generated about them. Greeks had small altars to gods in their own homes, and gave sacrifices regularly. They also often had statues to honor the gods in their homes.
The Greeks have some of the most famous literature to date. In fact, the Greeks invented drama. Many Greek stories come in the form of myths, tales about their gods. But also many plays were written, and epics as well. Religion plays a more important role in these than politics do, except in the case of comedies, that were often political satire. The best way to go about learning about how religion and politics influenced Greek literature is to read examples of such influenced literature:
Pieces of Greek Poetry - Comments on some Greek Tragedies - The Divine Beauty contest - Epics