Buttercream icing tastes best, but tends to be a bit yellow (from the butter and real vanilla); Royal Icing is more "white."
Don't use liquid food coloring -- it thins out the icing too much!
You need a gel or paste coloring agent -- Wilton makes good ones, and they're available at Michael's or AC Moore and probably at your local craft store.
Personally, I've had the best response to sugar cookies with buttercream frosting and colored sugars -- to get deep colors you need to use quite a bit of the coloring agent and it can leave a funny taste (in my experience).
Good luck!
Alicia