Liberate each sector of Mars. To liberate a sector, you need to:
Destroy all EDF structures, this will help bring EDF control down a lot faster. These are blue markers with a star in the middle on your map.
Complete the Story related mission (yellow diamond)
Completely drain EDF control. You do this by killing EDF soldiers & destroying EDF owned property (red dots on the map). It is not necessary to finish all of the Guerrilla Actions (green circles on your map) to liberate a sector, but it will work towards other achievements. You can do these after you finish the campaign.