When you have completed one mission, you should have the next one listed and selected as default when you select"continue campaign". It would really surprise me if this doesn't work correctly for everybody. Also, when you have completed tutorial 1, and then select "next mission" in the victory screen, you should automatically go to the next mission.
Note that all missions, including tutorials, can also be played as single games. Then they will not give achievements or count towards campaign completion. Only when you play tutorials 1-3 as part of the campaign will you advance through it (there are also 3 extra tutorials outside the campaign).
As for My Games, you should see a "Manage games" section near the top where you can select the game from a dropdown menu and enter your code (this is displayed in Steam).