In this chapter, Joseph interprets Pharaoh's dreams and is promoted to rule all of Egypt. A famine occurs, but Joseph has prepared for it in time.
The first thing we see is that two years have passed, so we know Joseph has been in the prison for some time. The prior chapter established two key facts: we saw that Joseph had a gift of interpreting dreams and the king's cupbearer also discovered this fact when he was reestablished in his prior position.
This leads us directly to the second thing, which is Pharaoh now having two dreams. Again, the two dreams here are a parallel of Joseph's two dreams at the beginning and the two dreams we saw in the last chapter. Just as Joseph interpreted the dreams of the cupbearer and the baker, he proves that he can interpret the dreams of the mighty Pharaoh himself.
We also see Joseph establish the supremacy of the LORD over the powers of the magicians and wise men. This is a common theme in the bible, repeated again in the story of Moses and later in the stories of Elijah and Jeremiah in different forms. The simplest explanation is that "interpretations belong to God" and therefore anyone not given a divine interpretation cannot properly interpret a dream. However, the stories of Moses, Elijah and Jeremiah do not relate to dream interpretation, so there is a more general message here: the LORD is supreme over human agency in all realms of wisdom, knowledge and power. Moses and Elijah have power and prayer confrontations with unbelievers and Jeremiah has a prophetic confrontation with a group of false prophets about what is the true word of the LORD.
However, the closest parallel to this story is the story of Daniel, where Nebuchadnezzar seeks an interpretation of his dreams and all of his wise men fail, but Daniel (through the power of the LORD) provides the correct interpretation and is granted power and authority in the Babylonian kingdom. This is almost exactly what happens here to Joseph, with a few key differences. First, the interpretation of the dream is markedly different. Second, Pharaoh does not show the same cynicism towards his magicians that Nebuchadnezzar has. In the story of Daniel, Nebuchadnezzar refuses to even tell others his dreams, because he thinks they will contrive an answer to please him, but that is false. In this case, Pharaoh has no such fears and widely shares his dreams in search of an answer.
All of this seems pretty didactic to me, so I don't really want to spend much time on it. In fact, most of the story of Joseph's life seems didactic, in terms of its general structure and message. I suppose it could be interesting for people who are new to the bible, but in literary or theological terms it is not as interesting to me as e.g. the life of Jacob, which has richer character portrayals.
Some minor points:
Every time I read this I am amused when it says that they shaved Joseph and put him in new clothes. It's a brief addition, but it amuses me because you see how they try to control the appearance of people who are brought before Pharaoh. Even prisoners are made to not look like they just came out of a dungeon.
This chapter is where the "two dreams means it's serious" principle, which Joseph states in his interpretation. It is first stated here, but it is used earlier when Joseph has two dreams with the same interpretation.
In my opinion, verse 38 is one of the most important in this chapter. Pharaoh declares he wants someone "in whom is a divine spirit", and then chooses Joseph. We already know that Joseph is sharing interpretations which "belong to God", and now we have a strong implication from Pharaoh that he considers Joseph to be a container of divinity. In my opinion this is one of the earliest statements about the indwelling of the Holy Spirit in the entire bible. Most people (when looking for the earliest statement) usually go to the book of Exodus and Bezalel (Exodus 31:1-6), but I've always considered Joseph to really be the first. Of course, one can speculate about even earlier figures, such as Enoch or perhaps even Adam (in whom God blew his breath/spirit). But Joseph is the first person who shows divine inspiration in interpreting dreams and is the first person ascribed with a divine spirit. Note that the term translated "divine spirit" by the NASB is the Hebrew "ruach elohiym", literally "the spirit of God". The only reason I can think of why it is translated abnormally here is that the NASB translators considered it improper for Pharaoh to be speaking about God when he is not part of the Hebrew faith. The underlying text is precisely speaking about the spirit of God.
It is a peculiar and striking omission that nowhere in the entire story of Joseph's life has God spoken directly to him. Even when interpreting dreams, Joseph always interprets them correctly without being explicitly given the interpretation by the LORD. While it does say that the LORD gave Joseph favor with both the captain of the guard and the chief jailer, we have seen nothing of Joseph's direct relationship with the LORD. This is a strong contrast to the patriarchs, Abraham, Isaac and Jacob who were regularly instructed by the LORD. I don't know if there is a reason for this, but if anything it makes God's intervention in his life all the more poignant. So much of Joseph's life is centered around his relations with other people, who are generally his masters (first his father, then the captain, then the jailer, and now Pharaoh), yet through the whole process we have seen divine protection and favor in all of these situations.
The word in verse 43, in the NASB translated as "Bow the knee!" is the Hebrew Abrech (or Abreck), and it occurs only here in the entire bible. The exact meaning is broadly debated based on what it might be derived from. Regardless, from the context it is clearly a reference to Joseph's authority and the submission demanded from all of the commoners.
Joseph is given an Egyptian wife. While it is pretty clearly bad for the Israelites to marry Canaanites, I'm not sure if it's actually taboo to marry Egyptians. Moses, for instance, marries a Midianite, who are not sons of Jacob/Israel. And as with Esau, Joseph is not actually under the Mosaic Covenant, so these restrictions are still generally cultural and not legal or religious.
Joseph has two sons and there is no sign of the hereditary barrenness of his forefathers. From now on, the Israelites will multiply rapidly and become very numerous.
We also see another famine. This is not the first famine in the bible, as there were also famines during Abraham's life and during Isaac's. As I said back then, famines happened often in this geography (though remember, I'm only listing one famine per generation so there is still a 20-50 year gap between major famines).
Lastly, we see that Joseph's solution of gathering food during the abundance is going to save many lives now that the famine had come. There would have been no natural prediction of the famine after 7 years of abundance, but thanks to Joseph's foresight many will now live.