The book opens with Creation. God doesn't fashion the world with his hands, instead he speaks and whatever he says happens. On the first day he makes light by saying “let there be light,” but he doesn't make the sun, moon, or stars until the fourth day, which makes this command even cooler, since the light is compelled to exist without an apparent source. Then he makes the sky, sea, land, plants, animals, and man on various days and rests on the seventh day of creation. This seventh-day rest or sabbath is a symbolic thing—obviously, God is not fatigued—and it becomes an important part of Judaism later on (less so for Christianity, depending on who you ask).
God says that everything he's made is good, so naturally, trouble is just around the corner. Adam and Eve are innocent and haven't done anything wrong yet, but they have free will, aka, the ability to do wrong. They eat fruit from the one tree forbidden to them, and for that transgression, they become subject to death and grow aware of moral taboos. In literature, a change of clothes denotes a change of station, but when Adam and Eve put on their animal-skin clothes, it's showing a negative change—they have definitely come down in the world, and they are sent away from Eden.
Then we get Cain and Abel, Noah and the Flood, the Tower of Babel, and a lot of genealogy in quick succession, but most of Genesis is a march toward Abraham, and most of what happens after Abraham is a march toward Moses in the next book. In chapter 12, God tells Abram to leave his home country and go to a new place. Abram's children and grandchildren are going to become a major nation, and they're going to be blessed and cared for by God himself. This is where, to me, things really get going. God is establishing his people-group, who are eventually known as the Israelites, the Hebrews, and the Jewish nation. God's not just intervening to save one person and his family, like with Noah, he's actively setting up an ongoing relationship with humanity through Abraham and his descendants.
It's interesting that God's people aren't goody-goodys. Sometimes they're downright awful. Even Abraham messed up a few times--he lied about being married to Sarah (twice!) and failed to trust God when he said that Abraham's heir would come through Sarah. His son Isaac pulls the same trick of lying about his wife, and Isaac also favors one of his children over the other, which has some very negative effects. Isaac's son Jacob makes even bigger mistakes when he tricks his brother and fools his father, but he has to pay for his deceit by having to marry a woman he doesn't want and having to work for a father-in-law who cheats him out of his wages for a couple of decades. And as far as romantic connections go--I always notice them, even when reading scripture--the couples are mainly established to produce descendants, so you end up with some odd combinations like Abraham and Sarah, who were half-siblings (which was okay back then) and Jacob and Leah-Rachel-Bilhah-Zilpah, which is presented as a “this was legally permissible, but nobody was happy with it” arrangement. After reading Genesis, the chief feeling I come away with is deep relief that I didn't live back then. Nobody had it good, but the people who were part of God's covenant at least knew that they had an ultimate purpose.
Jacob's son Joseph is really a stand-out character to me. He seems to do less wrong than anybody else, and he accomplishes more obviously helpful and virtuous tasks than most of his immediate ancestors. He refuses to sleep with his boss' wife even when she's hunting him down for that purpose, he rises from being a lowly prisoner to the second most powerful man in Egypt, he forgives his ten older brothers for selling him into slavery, and he trusts God and saves his family and his new country from a seven-year famine. Joseph is just head and shoulders above his brothers in terms of goodness, and we aren't shown that he does anything wrong at all. His whole extended family moves to Egypt with him to sit out the famine, and that's the setup for Exodus, the book where the Moses comes to deliver God's people from slavery.
Note: I'm a Christian, so I do believe that all of this happened. This isn't a book review, just a quick catalogue of my thoughts on books of the Bible.