These are the top Minneapolis area museums, however, for a much bigger list of free museums in Minneapolis check out this journal post;

Walker Art Center Top ten in the world modern art museum

Minneapolis Institute of Art (MIA) huge free museum of every kind of art

Bakken Museum young adults science museum focused on electricity

Russian Museum well curated fluid museum

Swedish Institute monstrous mansion that also houses rotating art exhibits

Mill City Museum best history museum in the Twin Cities