From John MacArthur’s Bible notes:
2:3, 4 The careful actions of Moses’ mother to construct the ark of bulrushes, to set Moses afloat close to the royal bathing place, and to have his sister watch to see what would happen, indicate a hope that something would work out right for the child.
2:5 the daughter of Pharaoh. She has been identified, possibly, as Hatshepsut or another princess; in either case, she was a princess whom God providentially used to override Pharaoh’s death decree and protect the life of His chosen leader for the Israelites.
2:10 became her son. The position of “son” undoubtedly granted Moses special privileges belonging to nobility, but none of these persuaded Moses to relinquish his native origin. Rather, as the NT advises, his spiritual maturity was such that when he came of age, “he refused to be called the son of Pharaoh’s daughter” (Heb. 11:24). The formal education in the court of that time meant that Moses would have learned reading, writing, arithmetic, and perhaps one or more of the languages of Canaan. He would also have participated in various outdoor sports, e.g., archery and horseback riding, two favorites of the Eighteenth Dynasty court.
2:11 when Moses was grown. The narrative skips over all details of Moses’ life as the adopted son of an Egyptian princess prior to the event which led to his flight into Midian.