Funny how condensed the Bible can be when it wants to. Modern retellings of Exodus (like The Prince of Egypt) focus a lot on Moses's early years. The actual text, though, only gives this period of Moses's life one measly chapter. He goes from birth to married in a mere 25 verses. Keep your eyes on this guy.
So here's how it goes down.
A boy is born to a Levite Hebrew woman who hides him for three months—remember the whole kill-the-firstborn thing? Then she puts him in a basket in the reeds of the Nile, and asks Miriam (the boy's sister) to keep an eye on him.
Meanwhile, Pharaoh's daughter is taking a bath in the river. She finds the baby and thinks he's cute. She wants to keep him after recognizing that he must be one of these Hebrew boys.
Conveniently, Miriam shows up and says, "I can find you a woman to breast feed this kid!" So Miriam runs and gets the baby's mother to do the job.
The kid grows up past breastfeeding age, and Pharaoh's daughter takes him as her son. She calls him Moses, which means "I took him from the water." Moses is an Egyptian name, incidentally.
Moses grows up knowing he's a Hebrew in Pharaoh's household. One day, he goes for a jaunty stroll to see his Hebrew brethren and sees an Egyptian beating a Hebrew. He makes sure no one's around, kills the Egyptian, and leaves.
A quick note: "Hebrews" and "Israelites" mean the same thing. "Hebrews" comes from the word for the Hebrew language, and "Israelites" comes from Israel, the land of the Israelites. Today, an equivalent of this might be "English-speaker" versus "American" or "British"; but for our purposes, "Hebrews" and "Israelites" refer to the same group.
Later, Moses sees two Hebrews fighting and breaks it up. The Hebrew yells, "Who made you a judge and ruler over us? Do you mean to kill me as you killed the Egyptian?" (2:14)
Moses freaks out and runs away.
Pharaoh finds out.
Moses settles in Midian and helps out some Midian women trying to tend their father Reuel's flock.
The women tell Reuel about the help Moses gave them, and Reuel gives Moses one of his daughters, Zipporah, as a thank you. Yeah.
They have a son and name him Gershom.
God is still around—he remembers his covenant with the Israelites and hears their cries.