How important were the Greek gods to the citizens and what role did religion play in ancient Greek civilization?

Answer 1

Greek life centered around religion, and every problem had a god or goddess to fix it. The great majority of the Greek oral tradition was about the gods. The intricate mythologies of the Greeks were created by the human desire to better understand the surrounding world. Example: the idea that Zeus was responsible for lightning. The Greeks disn't have wikipedia so a lot of what they didn't understand became explainable by the gods.

Addition:

There were two aspects - the mystery cults and the general religion.

The cults were generally carryovers of more primitive religious practices, associated with either an older earth-goddess (eg Demeter/Persephone) or a later sky-god (eg Appollo). The oracle of Delphi was a fusion of both earth and sky gods. These cults often offered the keys to an after-life, but were expensive and restricted to a comparatively few initiates. From these and others in the Middle East arose such later mystery cults as Mithraism, Isis and Christianity, which travelled from their original cult centres and so became widely accessible.

The general religion was based on sky gods, which is natural as they came with the nomadic Greeks when they took over from the earlier agricultural peoples and their earth gods. As with the Roman official religion, it was seen as a compact between the state and the gods. The gods were given their due (respect, consultation and sacrifice) and it was hoped that in return they would influence events or give warnings to the benefit of the state and its people. However underlying this was a cynical use of religious practices by leaders to control the people: the priests were of the aristocracy. Nothing has changed today.

In consequence, all significant acts and many personal acts were accompanied by getting oracles, sacrificing and invoking divine intervention or approval. Even the games and theatric performances were part of religious festivals dedicated to gods.

While they had a somewhat ambivalent attitude which could almost be classed as doubt, it was in fact double-think. While they may have made jokes at the gods' expense (buzzing like blowfiles above a sacrifice), they also were meticulous in their practices, and assigned the death penalty for sacrilege (exit Socrates). In an uncertain world of which they knew relatively nothing of physical cause and effect, the gods and myths provided answers of the unknowable, non-understandable and unpredictable.

The fact that so many today, in a world where scientific knowledge explains all this, still cling to extreme religious beliefs is an indication that superstition was and is hard-wired into our minds as an evolutionary advantage. The Greeks did what so many do today, just with differently-named gods and rituals.