Based on all the images and information released thus far, here is, what I think, is the best working theory for the different colored icons on the Character cards. Let's take them in order that they appear.

White = Recruitment. This is the most controversial suggestion, but note the shape of the blur behind the arrow, it looks like the shape of a character standee. Even more conclusive are the starting four mission cards down in the Lefthand corner of the overview image. Four mission cards, each with a return to hand symbol, each featuring a different color symbol. Only four of the 12 released characters have a white icon, 2 apiece on starting characters. I would be surprised if any other white icons appear. Perhaps specific recruitment missions will appear (to get Luke or Boba Fett out of the pile).

Blue = Espionage. Blue appears on the probe droid mission and the homing beacon mission.

Gold = Influence. Used for terror or diplomacy

Red = Combat. Obvious.

Any other thoughts?