The river, and the palace
Pharaoh was killing the sons of the Children of Israel. Musa's mother was inspired to place him in the river, and the Quran records the promise made to her: Do not fear and do not grieve; indeed, We will return him to you and will make him one of the messengers (28:7).
He was picked up by Pharaoh's own household — so that he would become to them an enemy and a source of grief (28:8). And then, because the baby would not accept any nurse, his sister suggested a household that could care for him, and he was returned to his own mother, who was then paid to nurse her own son (28:12-13). The Quran says plainly why: that We might console her.
Children love this part, and it is worth telling slowly. The plan to destroy him was the exact route by which he was saved and returned home.
Madyan
As a grown man he intervened in a fight and struck a man who died. The Quran records his immediate reaction: This is from the work of Satan, and then My Lord, indeed I have wronged myself, so forgive me (28:15-16). He was forgiven, and he left the city afraid.
In Madyan he helped two women water their flock and then sat down and said one of the most quietly moving du'as in the Quran: My Lord, indeed I am, for whatever good You would send down to me, in need (28:24). He had nothing, and he did not specify.
One of the women came back and offered him work, and he stayed there years.
The fire, and the du'a
Travelling with his family he saw a fire on the side of a mountain and went to fetch a burning branch. What he found instead was being addressed: Indeed, I am your Lord, so remove your sandals. Indeed, you are in the sacred valley of Tuwa (20:11-12).
He was given the staff and the hand as signs, and told to go to Pharaoh. And then the passage that matters most for a child: he asked for help. My Lord, expand for me my breast, and ease for me my task, and untie the knot from my tongue that they may understand my speech (20:25-28). He also asked for his brother: And appoint for me a minister from my family — Harun, my brother (20:29-30).
He said he was afraid (20:45). He said he was not the better speaker (28:34). He asked for his brother to come with him. And he was sent anyway, with his request granted: You have been granted your request, O Musa (20:36).
Pharaoh, the magicians and the sea
He was told to speak to Pharaoh with gentle speech, that perhaps he may be reminded or fear (20:44) — which is a startling instruction given who Pharaoh was, and worth pointing out.
The magicians were gathered, and their ropes and staffs appeared to move. Musa's staff swallowed what they had made. And then the magicians — the professionals, the ones who knew exactly how the trick was done — fell down in prostration and said they believed, and would not take it back even when Pharaoh threatened them (20:70-73). That is the real turn in the story.
At the sea, with the army behind them, his people said they were caught. His answer: No! Indeed, with me is my Lord; He will guide me (26:62). The sea parted, they crossed, and Pharaoh did not.
Muslims still remember that day every year. It is Ashura, the 10th of Muharram, and the Prophet صلى الله عليه وسلم fasted it because Allah saved Musa on it (Sahih al-Bukhari and Sahih Muslim) — more on our page on the Islamic months for kids.