I would suggest that for cheap flights you are better booking in advance. If you look at the
Flight comparison tool you can check the flights to Minnesota. It will compare flights from several airlines and show you the cheapest fares available for the dates you wish to travel.
Since it is backpacking you are doing it would be helpful to look at the
transportation section. Here you will find the best ways to travel around Minnesota to each destination.
I think it would be best to have an idea of where in Minnesota you would like to visit before you set off on your trip. If you look at the different areas of
Minnesota and look at the
things to do you could plan an itinerary.