Aaron was Moses' older brother (by three years, Ex. 7:7). Aaron married Elisheba and had four sons. He and Moses "sprang" not from the firstborn (Reuben), but from Levi, Jacob's third son. Their father, Amram, was the firstborn of Levi's second son. This all suggests that God's election of Moses and Aaron was not based on any sense of inheritance or privilege. Rather God chose them out of His grace and will.
The first mention of Aaron occurs at Ex. 4:14. Moses was resisting God's directive to bring His people out of Egypt. God offered to send Aaron to help him because Aaron could "speak well."
God would be with all their words and teach them what to say -- illustrating a very close relationship between God and his messenger.
Aaron was already on his way to meet Moses in Midian when God chose him to be Moses' helper. Perhaps Aaron intended to tell him that Pharaoh had died and that it was safe for him to return to Egypt.
Aaron was very happy when he saw Moses in Midian, and he kissed him.
Aaron performed the signs before the elders once they had returned to Egypt.
Moses and Aaron were quite the team -- Moses was like a god to Pharaoh (speaking with authority); Aaron was like a prophet (addressing the people with the words he was told to speak)
Both Moses and Aaron went to the first meeting with Pharaoh. It did not go well and resulted in the loss of straw for making bricks. The Israelites were furious with Moses and Aaron when they found out the reason for Pharaoh's order.
Scholars really aren't sure how to explain the story whereby Aaron's rod became a serpent. When Pharaoh's magicians repeated the act, Aaron's serpent swallowed up the other serpents.
Moses and Aaron stood shoulder to shoulder against Pharaoh throughout all the plagues, the Passover instructions, and the march out of Egypt.
After the exodus, Aaron's next big job was to call all the people together and help Moses explain about the manna and quail.
Aaron (and Hur) held up Moses' hands during the battle with the Amalekites. When Moses' hands were up, the Israelites were winning; when the hands came down, the Amalekites started winning. They held up Moses' hands until sunset. Joshua overcame the Amalekites.
Aaron was one of the elders who ate bread with Jethro (Moses' father-in-law) in the presence of God. (Ex 18:12)
Just prior to being given the Ten Commandments, God invited Moses to bring Aaron with him up the mountain. Aaron is not mentioned again for many chapters, but does this mean that he was there the whole time? (See Ex 19:24)
After the covenant had been ratified, Moses, Aaron, his two sons, and seventy elders were invited up the mountain to worship God. They were allowed to "see" God, without harm to themselves. Then Moses and Joshua went farther up -- Moses to go into the cloud (for 40 days and nights), Joshua to wait patiently.
While on the mountaintop, Moses was given his first instructions regarding the Aaronic priesthood, including duties (keep the lamps burning from morning till night) and garments to wear. This is followed by detailed instructions regarding their ordination ceremony. (The idea behind the detailed clothing requirements emphasized the office, not the person.)
Aaron was to be the high priest; his sons were to be the ministering priests. They were to be consecrated and pure, so the ceremony included actual washing. They also sacrificed bulls for any sins they might have committed. (See Ex. 28-30)
At precisely the same time Moses and God were having this exalted conversation about Aaron, he was fulfilling the Israelites' request to make "gods who will go before us." (Showing that even the holiest of men can be persuaded to do what is contrary to his beliefs.)