In this chapter, Moses is born, adopted by Pharaoh's daughter, and exiled to Midian.
This story, beginning with the birth of Moses and ending with the crossing of the Red Sea, has to be one of the best known stories in the entire bible, and it's appropriately so because it is genuinely foundational to many of the tropes that we will observe in the rest of the OT and the NT. I already mentioned a few of these in the last chapter, such as the "baptism in the Red Sea" and Moses as a type of Christ, but there are many more.
Firstly, note that Moses is a descendant of Levi. Remember from Genesis 49 when I said the King would descend from Judah? And King David is a son of Judah, and Jesus is a descendant of King David. This means that Moses and Jesus come from different tribes.
But I also said that Moses is a type of Christ. An astute reader would criticize me here, because if these two men descend from different tribes, how can one be a type of the other? And this is a fair criticism, because as we saw in Genesis 49, each of the different tribes has its own prophetic destiny, which is stated partially by Jacob. In that sense, the distinction between the tribes is at least as important as the union of the tribes under Israel, that is Jacob.
Edit: changed my mind about the paragraph below and altered it moderately.
Nevertheless, I insist that Moses is a type of Christ, and now I'll explain why as briefly as I can, even though this is jumping ahead. There are three leadership roles of importance in the OT: the prophet, the priest and the king. Judah is the household of the king, that is established in Genesis 49. What we will see here is that Levi is the household of the priest, but despite this Moses holds the role of the prophet. The priestly line will emerge shortly from Moses's own household and from his brother, Aaron. These three strains are always separated in the OT into distinct roles with distinct personages holding each given position. This is almost certainly deliberate, as the three functions act as checks and balances against each other, much like a modern parliamentary government. In the NT, we will see Jesus hold and be described as fulfilling all of the three roles, being a prophet (Revelation 19:10), a priest (Hebrews 5:1-10), and a king (various, such as John 19:19-22). This is one of the many cool things about the bible, where the disparate roles in the OT are brought into a unity in Christ.
What this means is that Moses is a type of Christ, but he's not the only one. There are many types of Christ in the OT. Another one that we have already seen is Isaac, who was to be sacrificed by his father Abraham in Genesis 22. This is a figurative parallel for Christ, who was sacrificed by his Father God to bring the forgiveness of sin.
Moving on, next we see that Moses is spared from death by the actions of his mother, and through a series of coincidences, he is adopted by Pharaoh's daughter, who has pity on him. It is certainly ironic that the adopted son of Pharaoh, who instituted this murder policy, would be the very agent of the Israelite's freedom and Pharaoh's ruin. It is interesting that Moses would be in Pharaoh's household at all. As the reader can note, we are told almost nothing about Moses's childhood and youth, up until the time that he is old and strong enough to kill another man. What we do know is that he spent all this time as the adopted grandson of Pharaoh, so he would certainly have been well-educated and pampered by the wealth of his household.
And the next thing that is spoken of Moses is that he kills an Egyptian for abusing a Hebrew. This is described so briefly, yet it speaks so much. We can infer from this a lot of things: Moses knew he was Hebrew (how did he know? He was raised since childhood in Pharaoh's household. He must have been told by his mother or sister, but we never see this happen). It says that Moses looked upon the hard labor of his brethren, so we know that Moses is concerned with their fate. And then he kills an Egyptian who was abusing one of his brethren, showing that not only was Moses concerned with their fate, but he had an obvious passionate desire to take personal action to help the Hebrews in oppression. Again, I can only ponder how he was raised such that he's willing to risk his own life to help his people, when he was not raised by his people. Just as much as one wonders how he became like this, one also wonders how this passionate defense of the Hebrews and Moses's commitment to justice in general impacts his later life as the leader of the Israelite people. I have heard some commentators suggest that it is critically indicative of his future life.
Regardless of that, we can see the immediate impact here. Pharaoh discovers this and attempts to kill Moses in retaliation. Moses is forced to flee into the wilderness of Midian. While the exact location of this land is disputed, the Midianite people were highly nomadic shepherds, so it would have been an arid and uncultivated land, much like the origins of Abraham's family and very unlike the rich courts of Cairo. And thus, Moses chose to risk foresaking the wealth of Egypt out of his desire to free the Israelites: while Moses did attempt to hide his action, he would have known that he was risking being discovered, and in fact that is what happened.
In the very next scene, we see Moses defends the daughters of Reuel, the priest of Midian, from the shepherds who we can presume are just trying to protect their well. As we have seen before, water is almost always a source of contention in these arid lands, as even wells can run dry if they are overtapped. However, Moses remains committed to justice and protecting the relatively defenseless women. Moses then marries a Midianite, which is probably OK because Midianites are not a people of Canaan.
Lastly, it says that the Israelites groaned in their bondage, and that God heard them. This is obviously a setup for what happens next, the burning bush and the commissioning of Moses. But it's very interesting for us, as readers, because we have seen Moses being prepared to rescue the Israelites and Moses's life for the past two chapters. We have seen his miraculous salvation from death when he was born, we have seen his passion for justice and the freedom of his people, and we have seen his long desolate exile in Midian (he was in Midian long enough to get married and have a son, as well as the king of Egypt dying, so he was in Midian for at least 2-5 years, possibly much longer). It was so desolate he named his son, Gershom, Stranger there. In contemporary English, "A Foreigner Living in a Foreign Land". All of these things, his rich childhood and his desolate adulthood, groomed him for his tasks ahead, about which we now hear the crying of the Israelites and the impending action of the LORD. One could argue that verses 23-25 actually describe the outcry of the Israelites over many years, its position here is clearly linked with what follows, the commissioning of Moses to go and free the people and the plagues that are to follow. I think this is really cool because in spite of the immediacy of the burning bush event, we know that there were literally decades of events that preceded this outcry and divine response, during which Moses was prepared to fulfill God's commands and answer the prayers of the Israelites in bondage. This is a symbolic picture of many such prayers, which may have a sense of immediacy to us in our prayers and in God's response, yet so often there are preceding decades of groundwork being laid for God's response, that even precedes our prayers. Or in simpler terms, God is working to answer our prayers even before we pray.
One last minor note: we see a reference to the covenant with Abraham, Isaac and Jacob. This is a very common trope in the OT. God's remembrance of the covenant is a driver for many of his actions, as it is here.