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The head of the screw is small enough (you did get the trim head screws?) that you don't need to countersink.
If you are going to paint then I'd use spackle or putty. Don't forget the primer.
If you are going to stain them, do the staining first. Then get colored putty to fill the holes. You can mix different colors to match your stain. I did that and one has to look really hard to see where the screws are located.