As I previously mentioned, Joseph's life is largely dominated by the two dreams he has, and the subsequent tribulations. He has many ups and downs. His first "up" was the favoritism of his father, then he had the down of imprisonment and sale into slavery. Then he has the "up" of favor with his master, Potiphar, then the down of being falsely accused of rape and sent into prison. But then he has the "up" of favor with the chief jailer.
As of this chapter, it's still an open question whether he will be embittered by the experience. We will later discover that he is not.
The first thing we see, after Joseph is taken into slavery, is that he is gifted by the LORD and prospers in everything he does. It states that Potiphar "saw that the LORD was with him" and therefore entrusts him with everything. This does not mean that Potiphar acknowledges the LORD as God, but it is perhaps a general expression of favor to say "the LORD is with
Either way, Joseph is clearly gifted and favored by God, which is interesting because so far, we have no seen God directly interact with Joseph or speak to him in any direct way. Joseph had the two dreams, but neither dream mentions God in any way, they simply reflect Joseph's relationship to his family. With all of Abraham, Isaac and Jacob, the promise of God was directly stated to them by the Lord at one point or another. With Joseph, this has not happened, yet he is gifted and favored, and he acts out with that gifting in a simple way.
In my opinion, Joseph is continuing with the same simplicity and innocence that we saw in chapter 37, in that he serves Potiphar without reservation. He refuses Potiphar's wife's advances, but there is no reference to him ever doing anything else. He never shows any of the deceptiveness that runs in his family, which we have seen in Jacob and in all of his other sons.
We know from chapter 38 that Joseph is in Egypt for a very long time, since Judah was separated from his brothers for 20-40 years. Nevertheless, there are no explicit dates or times given for Joseph's various journeys, so one can only guess how long he was in any specific phase (Potiphar's house, jail, later stuff). The story is very compressed, which makes it feel really short, but it is not actually short at all. This chapter describes Joseph's entire time in Potiphar's house, but he was possibly there for over 10 years.
Why did Potiphar's wife accuse Joseph of rape? The text doesn't say, so I wonder about this. The most likely explanation I know of is that she felt ashamed or angry at his refusals.
It's very interesting that Joseph is sent to the royal prison. This shows very clearly that Potiphar did not entirely trust his wife's account, because the normal punishment would have been death. Not only was Joseph sent into prison, but he was sent into the royal prison "where the king's prisoners were confined". This would be the nicest of the prisons, because political prisoners are usually more important than common criminals and they are not always going to be killed.
I also think it's interesting how closely the account of Joseph's time in the prison parallels his time under Potiphar. In both cases, he ascends to the rank of chief servant, and everything is put under his care such that his master did not pay attention to anything he did because everything he did went well. We have seen similar patterns of success under Jacob, Isaac and Abraham, but in all of those cases they were working for their own profit rather than another (the semi-exception to this being Jacob's complicated relationship with Laban where he traded his labor for wives or wages respectively). Even so, there is definitely a continuity here. Joseph is a sort of spiritual successor to Jacob, even though he is not given the divine promise by God, and he is not an ancestor to the Messiah.
This success is juxtaposed with the "bad report" that we have about Jacob's other sons, who are perhaps less diligent about their shepherding tasks. These other sons are clearly more combative (arguing amongst each other, showing their hatred to Joseph, killing the Shechemites, etc) and less diligent than Joseph. Jacob probably sees more of himself in Joseph than his other sons, and that might be why he favors Joseph above his other sons.