How did Moses find out he was adopted?

Answer 1

1) In those days, Israelite custom was to nurse babies for up to four years. Since it was his own mother who nursed him (Exodus 2:8-9), his family had plenty of time to teach him who he was before he was returned to Pharaoh's daughter (Exodus 2:10) to live in the palace.
2) Moses was not a prisoner in the royal palace. He came and went as he pleased (Exodus 2:11 and 2:13) and sought out his people (ibid).

More about Moses:

Moses was an Israelite, a great-great grandson of Jacob. He was born 245 years after the death of Abraham. The time when Moses was born was when the Pharaoh had ordered his people to kill all Israelite male infants because he (Pharaoh) was afraid that the Israelites would become too strong for him (Exodus ch.1-2).

Moses' mother didn't want him to die. So she made a basket for him and put him in it to float in the Nile reeds. He was found by Pharaoh's daughter, who took pity on him (Exodus ch.2) and raised him as her own son.

Moses was forced to flee after killing a cruel Egyptian taskmaster, and went to Midian, where he wedded the daughter of Jethro.

He eventually achieved the highest level of prophecy (Deuteronomy ch.34) and was called upon by God (Exodus ch.3). He brought the Israelites out of Egyptian slavery (Exodus ch.12). He received the Torah from God (Exodus 24:12) and later recorded it in writing (Deuteronomy 31:24). He went up on Mount Sinai for 40 days and nights (Deuteronomy ch.9-10) and brought down the Two Stone Tablets with the Ten Commandments (Exodus 31:18). He brought the Israelites into the covenant with God (Exodus ch.19 and ch.24), and he oversaw the building of the Tabernacle (Exodus ch.35-40). He was the humblest of men and the greatest of prophets (Numbers ch.12).

See also:

What problems did Moses face and overcome?

and Jewish history timeline

Answer 2

Moses discovered that he was adopted by meeting his brother and sister, who were among the Hebrew slaves.