There is no ancient Greek word for religion in the modern sense, and no institution corresponding to a church. What existed were the customary honours owed to gods, performed by a household, a village, a kin group, a trade or a city. The term usually translated as piety, eusebeia, describes correct conduct toward gods, parents, the dead and guests, all in the same breath. Impiety was a matter of neglect or violation rather than of holding wrong opinions: the prosecution of Socrates in 399 BCE turned on charges of not acknowledging the gods the city acknowledged and of introducing new ones, which is an allegation about cult and civic loyalty rather than about theology.
The central act was sacrifice. An animal was led to the altar, killed, its thigh bones and fat burned for the god, and the remainder cooked and eaten by those present. Most meat eaten in a Greek city had been sacrificed, which makes the rite as much a distribution of food and standing as an offering. Alongside it ran libations, first fruits, incense, prayer and votive dedications, and at the other end of the scale enormous civic festivals: the Panathenaia at Athens, or the Dionysia at which tragedies were performed as part of the worship of Dionysus.
Priesthoods were generally not vocations. Many were annual offices, allotted or elected, occasionally sold, and often reserved to a particular family. The holder administered a sanctuary and its funds and performed the rites; there was no preaching, because there was no body of doctrine to preach. This is why the religion accommodated philosophers who argued in public that the gods were nothing like the poems described. Theology was not the point of the exercise, and getting it wrong was not in itself an offence.