Greek religion was a mixture of old Minoan beliefs, Central Asian gods that the Indo-Europeans brought to Greece, and West Asian ideas they got from their neighbors. Like all of those people, people in ancient Greece believed that there were invisible, powerful gods and spirits that could control what happened to you. Most people also thought that you could control those gods and spirits through sacrifice, prayer, and living a good life.
The most important spirits and gods were natural things like rain and how plants grew. If you controlled these successfully, you would have enough food, and if not, you might starve. People also tried to control other natural forces like earthquakes, volcanoes, ocean storms, and plagues. And they prayed to win battles and wars. Usually whole towns prayed and sacrificed together for these big things. When an earthquake or a plague hit a town, people thought it must be something the whole town had done wrong, or maybe something the rulers of the town had done wrong, like King Oedipus killing his father.
Most people thought it mattered how you behaved, too - good people had good things happen to them, and bad things happened to bad people. People told stories about the gods to teach children how to behave. Many stories warned against hubris - thinking you were better than the gods - and against greed and cheating.
To find out what you should do, a lot of people went to oracles - special people or places where the gods or spirits would answer your questions. There were expensive oracles for rich people and towns, and simple oracles for ordinary people. People asked what medicine a sick person should take, and whether this was a good time to start a new business or a war, and also philosophical questions like what was the most beautiful thing?
Sometime people who felt they needed a fresh start with the gods joined a mystery cult, where you did special rituals and sacrifices and tried to form a special, closer relationship with a particular god or spirit. For most people in ancient Greece, the gods were always around them, paying attention to everything they did, and an important part of success in life was keeping on the right side of the gods. In the Archaic period, most people didn't think much about life after death, but gradually people began to think about the afterlife more, until by the Hellenistic period, about 200 BC, a lot of people did think that good people went to heaven and bad people went to hell.
To find out more about Greek myths, check out these books from Amazon or from your library:
Greek Myths for Young Children, by Heather Amery (Usborne, 2000). My niece really likes this one. For younger children.
Greek Myths for Young Children, by Marcia Williams (1995). In comic-book format. My son loved it.
D'aulaire's Book of Greek Myths, by Edgar and Ingri D'Aulaire. For older kids.
Mythology by Lady Hestia Evans, Dugald A. Steer (Editor)(2007) - My daughter loved this one! A new book in the Fairyopolis, Dragonology, etc. series.
Pandora's Box: A Three-Dimensional Celebration of the Mythology of Ancient Greece, by Sara Maitland and Christos Kondeatis (1995). Not really about Pandora specifically, but a complex of stories, games, and puzzles about Greek mythology. All ages.
Greek Religion, by Walter Burkert (reprinted 1987). By a leading expert, for adults. He has sections on each of the Greek gods, and discusses the deeper meanings of the Greek myths, and their function in Greek society.
Teaching and Dramatizing Greek Myths, by Josephine Davidson (1990). Advice for middle school teachers on teaching Greek myths by having the students write and perform skits.