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How, exactly, do I transport my troops across water? What do I actually do?

Select your army and then click on a province to which you want it to move. That can be any province on the map, then your troops will move to that province. It doesn't matter if that province is over-sea or not.

Though you might want to first march your army to a coastal province and only then order it to cross a sea. Since else you the game will calculate the cost of sea-travel from your starting province to your destination province, eventhough you might just cross 3 sea-provinces.