What do books 1 through 12 of The Odyssey teach us about Ancient Greek religion and values?
Just to add to readerofbooks' answer above, I'll mention that atheism did exist in ancient times and in Ancient Greek culture. Though religion was a foundational pillar of most states that granted authority to the government, not all Greek thinkers embraced traditional or mythological understandings of the world. Many early Greek philosophers in the 6th and 5th centuries BCE worked to understand the causes of natural phenomena through means other than the mythology already in place. Democritus and other atomists tried to explain the world in terms of atoms, and dramatists' characters gave voice to contrary opinions in their plays. A number of famous Sophists are said to have expressed disbelief in (or at the very least uncertainty about) the existence of the gods, and Socrates himself was accused of atheism (although he may not have actually believed in it). So, while atheism was perhaps uncommon and often illegal (typically a capital crime), it did exist and the Ancient Greeks would have perfectly understood the concept.
This is a good question. Here are a list of points we can learn from Homer's Odyssey.
First, the Greeks believed in the gods. In other words, there was no such thing as atheism in the ancient world. This is one of the points that separate the modern and ancient worlds.
Second, the ancient Greeks were polytheistic. This means that the Greeks did not believe in one god like the Jews or Christians. Just think of the Olympian gods. There are also many lesser gods and goddesses and other divine and semi-divine figures.
Third, these gods are anthropomorphic, that is, they are very much like humans. They have emotions of love, jealously, anger, and the like. This is why certain goddesses and gods love one place over another - some side with Trojans; others side with the Greeks.
Fourth, the gods are able to mate with humans and interact with them. They sometimes disguise themselves and come down to speak and guide humans.