The first mention of the Egyptian slavery is Breishit 15:13, when God tells Avraham that "your offspring will be strangers in a land not theirs, and shall serve them 400 years".
This prediction is strange in the context it comes in. There is no clear background on what led to the decree. And following this brief mention, the issue of eventual slavery seems to disappear from the book. As I suggested, there seems to be an causual chain of events throughout Breishit, which inevitably leads to slavery in Egypt. But this chain is really an example of "how", not "why"; the "why" for slavery remains unclear.
Perhaps we can gain some insight by looking at the story which immediately follows Brit Bein Habetarim (Breishit 15, in which the promise of slavery is made). In chapter 16, Sarah is childless. She has Avraham take Hagar, an Egyptian woman, as a second wife. When Hagar becomes pregnant, Sarah becomes displeased with her and oppresses her. Nevertheless, Hagar goes on to gives birth to a son (Ishmael), while Sarah remains childless.
This reminds me of the Israelites' experience in Egypt, but with the ethnicities reversed. In that second story, it was the Egyptians who invited Jews into their country. When the Israelites began to reproduce abnormally fast, it worried the Egyptians, who tried to compensate by oppressing the Israelites. But that failed, as the Israelites only continued to reproduce.
Due to the multiple similarities between the stories, I wonder if Sarah sinned in her treatment of Hagar the Egyptian, and if the oppression of Sarah's descendants is a measure-for-measure punishment for her misdeed. There are probably additional reasons for the slavery, but this could be one significant contributing factor.
If so, this would provide a clear linkage between chapters 15 and 16. Excluding the Avimelech episodes in chapters 20 and 21 (which I'm not really sure the purpose of), chapters 15 and 16 are the only stories in Avraham's life which did not seem to be closely thematically interwoven with what comes before and after them. I think we have now identified the connection between them.
(In principle, there's no reason why "random details that had to be mentioned somewhere" couldn't be thrown it at some random point. But it's much cooler if you can justify the placement as well as the existence of each story.)
A new king arose over Egypt, who did not know Yosef. (1:8)
The simplest interpretation of this line is that the new king knew Yosef's legacy, but chose to ignore it.
But it's also possible that at some point Yosef really became unknown. The ancient Egyptians had a tendency, especially around the time period the Israelites were there, to erase all historical records of whatever was inconvenient to them. And Yosef, because of his "abominable" Hebrew ancestry and his (presumable controversial) political and economic policies, would seem to be a likely target for historical censorship.
[Yocheved] became pregnant and bore a son. She saw that he was good, and concealed him for three months. (2:2)
He was good - when he was born, the whole house filled with light. (Rashi)
2:2 is a difficult line - what does it mean for a baby to be "good", and why would a mother choose to save only "good" babies? A couple possibilities suggest themselves to me (and other people): perhaps the baby was healthy and likely to survive (thus worth endangering oneself in order to save), or else was well-behaved (thus unlikely to cry constantly and reveal himself to Egyptians).
Rashi seems to take a different approach: that due to Moshe's historical importance, unusual effort had to be put into saving him. In order to explain how Moshe's future role was recognized by Yocheved, Rashi says that a miraculous light filled his house. As Siftei Hachamim points out, there is a precedent for equating the phrase "it was good" with light, in the Breishit creation story.
Now, Breishit 1:4 is not the only place in the Torah where the phrase "it was good" ("ki tov") appears. So please indulge me as I compose a midrash of my own, which probably relies on the same thought process as the original. Elsewhere, we see restfulness equated with good: "vayar menucha ki tov". Therefore, when the Torah says that baby Moshe was "ki tov", it may be saying that he constantly slept all the time. Therefore he did very little crying, and hiding him did not endanger his mother's life.