It's hard to imagine that after the deliverance from Egypt and all the things God had done for the Israelites that they would doubt in Him. But they did. Moses was on the mountain for 40 days and that's how long it took for them to fall apart. It illustrates the need for daily continual interaction with God.
I searched up some information on the golden calf because it seems so silly for Aaron to submit so easily to the demands of the people and found this (on this website):
While Moses was up on the mountain receiving God’s laws, the people were getting anxious down on the plain. Moses spent forty days (Exodus 24:18) up on the mountain with God, and by the end of that time, the people were beginning to think Moses had died or left them. The people urged Aaron, their temporary leader, to make gods for them to follow. Since they were accustomed to having visual representations of gods, this was the natural (but sinful) result of their thinking. Aaron took their gold earrings, which they had brought from Egypt, and melted them down to make a golden idol. The idol he crafted for them was a calf, but Aaron maintained the name of the Lord in connection with it (Exodus 32:5). He was merging the pagan practices they were familiar with and the worship of the God they were just beginning to be re-acquainted with. Aaron called the people together and told them that the golden calf was the god who delivered them from Egypt. The people offered sacrifices and then engaged inpagan rituals, including orgies (Exodus 32:25) to worship this new god.
Why did Aaron do this? Scripture doesn’t give us the full answer, but we can put certain clues together and get a fairly good picture. First, the people’s long familiarity with idol worship would incline them to follow that method in the absence of clear direction otherwise. It is likely that the people had not yet received the commands against idol worship, since Moses was yet to come down with the tablets of the law. Second, they were already in the habit of merging their beliefs with those of the people around them, a practice that would continue to plague them throughout the kingdom years. Third, Aaron was faced with an unruly crowd that placed a demand on him. The solution of making an idol and calling it by God's name seemed fairly reasonable.
Why did he choose a calf/bull? His lame excuse to Moses—“It just came out of the fire like this!”Exodus 32:24)—was just a feeble attempt to dodge blame. He fashioned it with a graving tool (Exodus 32:4) and took great care to form it that way. Some have tried to show that the bull represented one of the gods of Egypt, but that doesn’t fit the text, because Aaron called a feast to the Lord (Yahweh) and said that it was the god(s) which brought them out of the land of Egypt. The bull was a symbol of strength and fertility, and the people were already familiar with bull gods from Egypt. Bulls were also typical animals of sacrifice, so to use their image as a symbol of the god being worshiped was a natural connection. Aaron’s bull was a mixture of the powerful God who delivered the people through mighty works and the pagan methods of worship that were borrowed from the people around them.
Even though there are reasonable explanations for why Aaron and the people began to worship the golden calf, those explanations do not excuse the sin. God certainly held the people accountable for their corruption (Exodus 32:7–10) and was ready to destroy them for their sin. Moses’ personal intercession on behalf of his people saved them. Moses indicated that Aaron at least should have known that his actions were sinful (Exodus 32:21) and didn’t let him off the hook. As with any other sin, the punishment is death, and the only proper response is repentance. Moses called for those who were on the Lord’s side to come stand with him (Exodus 32:26). The Levites stood with him and were commanded to go through the camp and kill anyone who persisted in the idolatry. Three thousand men were killed that day. The next day, Moses went up and confessed the people’s sins before God, asking for His forgiveness. God declared that the guilty ones would yet pay with their own deaths and be blotted out of His book. These were the same ones who, on the verge of entering the Promised Land, would deny God’s promises and be sent into the wilderness to die for their sins. Their children would be the ones to receive God’s promised blessings.
Their experiences are a lesson to us today. Even though we might justify our actions through reason or logic, if we are violating God’s clear commands, we are sinning against Him, and He will hold us accountable for those sins. God is not to be worshiped with images, because any image we make will draw more attention to the work of our hands than the God who made all things. Also, there is no way we can ever fully represent the holiness and awesomeness of God through an image. To attempt to do so will always fall short. On top of this, God is a spirit (John 4:24), and we cannot form an image of a spirit. We worship God by believing His Word, obeying it, and declaring His greatness to others
We sin. We can explain our reasons and we can try and make excuses for our behaviour, but we deserve punishment for our sin. There are consequences for sin.
Our actions indicate our response to our faith.
When we sin, it is an action of defiance against our faith. Faith without works is dead. I think we can also say that faith with unrepentant sin is also dead. Sin threatens to invade our thoughts and our actions and separates us from God. Just like the Israelites struggled with idol worship, we struggle with idol worship. It may not be a golden calf but it might be the struggle of how we use our time, or what we focus on, or a handful of other sin but we are just as lost as the Israelites. It is only through repentance that we can be right with God. We demonstrate this right relationship with God through our actions that demonstrate our relationship with God and through that bring others closer to God.