I think there's a combination of terminology and information misunderstanding going on here, so let me try and explain this at an appropriate level. First, the phrase "travel at light speed immediately" doesn't make much sense. In physics, there's not really any such thing as "immediately." "Immediately" is synonymous with "instantaneously," and there's nothing we've ever measured that we can call instantaneous communication. But, regardless of a lack of instantaneous anything, phrased this way, I hope it's clear that saying something moves instantaneously at a finite speed is nonsensical from a conceptual point of view. It's like saying something is moving at 5 m/s and 20 m/s at the same time; 5 just doesn't equal 20, no matter how you slice it.
When you talk about the velocity of gravity, you're talking about the speed at which the force carrier of gravity (or the spacetime disturbance) propagates outward. That value is c, as far as we can tell. When you start talking about 9.8 meters per second squared, you're talking about the acceleration due to gravity, which is not the same thing. How hard you push and something and how fast it moves are related, but they're not the same, right? It's the difference between velocity and acceleration.
Now, if something provides a continuous acceleration, the object that is accelerating will keep going faster and faster, approaching a velocity of c. It doesn't matter what provides the acceleration; could be gravity, could be a rocket booster with infinite fuel. The point is, the speed that gravity propagates has nothing to do with how hard it pulls on objects. Those are completely different properties that are unrelated.
Finally, the gravitational pull between all objects does respond at speed c. This includes the earth and moon. But just because gravity "gets from" the earth to the moon at a speed c, that doesn't mean it causes the moon to move toward the earth with velocity c. I hope that clears up some of the confusion.