By train. Transportation by rail from Quebec to Chicago costed about ten American dollars in 1869.
Read this article; Concerning Emigration?, posted in the Norwegian "Billed-Magazin", spring 1869. It's interesting reading, and decribes several routes to the west, and how to cross Lake Michigan.
There are many articles on this site about the voyage from Norway and to the final destinations in America after landing in Quebec including this one that describes getting to Chicago:
There are many articles on this site about the voyage from Norway and to the final destinations in America after landing in Quebec including this one that describes getting to Chicago: