deltaG is the same thing as gibbs free energy. I think a lot of people dont want to touch this because there are so many types of this... but I am assuming this is a general chemistry question not a biochem+ question. Remember:
delta G= -RTlnK
the two are related as you said in the first part, but you are misunderstanding the point by the second part. If the reaction is at equilibrium, the delta G will be 0 because there is no "driving force" for that reaction to reach equilibrium. Think of it like a ball in a parabolic valley. Think of the ball as the reaction quotient K (Keq is the reaction quotient at equilibrium). Think of the potential energy of the ball due to gravity as the gibbs free energy of the reaction. If the ball is already on level ground (at equilibrium by this analogy), the delta G will be zero and the ball will not roll. If the ball is not at equilibrium (part way up the valley) the deltaG will be negative, so the ball will roll spontaneously. Hope that helps and good luck.
Alex