Presumably right after the Pharaoh (king) had ordered the destruction of male Hebrew babies by any and all, a Levite woman (you remember Levi? Son of Jacob? Me neither) hid her boy baby for a while after it was born. When it became too much she decided to put it in a waterproofed basket in the Nile instead. Fend for yourself, baby!
The Pharaoh's daughter found the basket and felt sorry for the baby. At this point Moses' sister, who saw the whole thing go down, suggested that the princess get a Hebrew woman to nurse it for her, so she did: the very baby's mother, who was paid as a wet nurse to suckle her own child. Sounds a bit convenient to me, but okay. When he was weaned he became the princess' son, Moses.
He grew up in the palace, I guess. They don't go into detail. After he became a man, he watched his own people at work in their slavery. When he saw an Egyptian beating a Hebrew, and there was no one around to see, he killed the Egyptian and buried his body.
Moses' first act, as related by the author, is murder. And cover-up. Ooooo, Nixon can tell you what the cover-up gets you.
Indeed, later he tried to settle some kind of fist fight dispute between two Hebrews and they taunted him by asking something like, "What are you gonna kill me like that Egyptian, huh?"
The Pharaoh tried to kill Moses, then. Personally? Maybe he hired some goons. I dunno. At any rate, like many a murderer, Moses went into hiding.
He ran to Midian, where there was a well. At the well he witnessed seven women trying to gather water for their father and being turned away by the shepherds there. So Moses stepped up and got them the water and sent them on their way. That was nice of him.
The women got home from this chore earlier than usual, so their father, the priest, asked them about it. "Oh, some Egyptian dude helped us out." What? Invite him for dinner!
Dinner turned into a betrothal. He married one of the daughters, Zipporah, and had a son, Gershom.
Meanwhile, back at the ranch, the king of Egypt died and the Israelites groaned and moaned about their slavery until their god became concerned.