Genesis and Exodus connect in the following ways:
As with all the books in the Bible there are multiple linkages between Genesis and Exodus. The most obvious linkage is normally that occuring in the immediate context shared between the books.
When surveying the end of Genesis and beginning of Exodus the death of Joseph is the obvious linkage between the books. Genesis tells the short story of Joseph's death and Exodus restates the event before moving several centuries forward in time to focus on Moses.
It's interesting to note what Joseph told his brothers before passing away.
Joseph reminded his brothers that some day in the future God would deliver their people from Egypt and they would return to the land of Abraham. Joseph then made them swear to carry his bones out of Egypt when that time came.
Joseph's reminder that God would one day bring Israel out of Egypt is a perfect intro to the events of Exodus. And of course, when that time came, the Israelites brought Joseph's bones with them.
There are other linkages shared between Genesis and Exodus. In one of Abraham's encounters with Yahvah he was told that his descendants would be slaves in another country for 400 years.
The 400 years of servitude is confirmed with a reference in Exodus to 430 total years spent in Egypt.
The difference in time of 30 years represents the time Israel spent in Egypt before being subjected to slavery. 30 years living in Egypt plus 400 years as slaves in Egypt equals 430 total years in Egypt.
The time in Egypt is a linkage between Genesis and Exodus because the crisis of being in Egypt begins in Genesis and the resolution comes by leaving Egypt in Exodus.
The promise given to Abraham about his descendants receiving the land of Canaan is also important to the basic storyline of Exodus. Here is one occurence of the land promise to Abraham.
The importance of the covenant with Abraham is obvious when reading through the book of Exodus. Often the stated reason for God's delivering the Israelites is in fulfillment of his promise and/or covenant with Abraham, Isaac and Jacob. Take for example the time Moses was told by Yahvah to assemble the elders of Israel.
When looking at these stories it's clear that God sees into the future. He promised land to Abraham's descendants, but explained they would be slaves in another country for 400 years. He then brought them out of Egypt in the book of Exodus in fulfillment of his promises to Abraham and the Israelites brought Joseph's bones as he had requested before his death.