Yes, Rivka, you are right that if you were sitting at the very center of the Earth, the Earth's gravity wouldn't try to move you away from your location, and so it wouldn't pull at you. If we define "weight" as the strength of the force with which gravity is pulling at you to change your location, then it is clear that you wouldn't weigh anything at all at the center of the Earth.
By the same criterion, you would weigh absolutely nothing at the center of a black hole. You would, however, be squished to a mathematical point. Even sitting at the center of the Earth, you would, for all your weightlessness, feel very heavy, compressed and trapped, with the entire mass of the Earth pressing in on you from all directions!
So, Rivka, you would weigh nothing at the exact center at the Earth. Is it at least theoretically possible, then, that if you were sitting an inch away from the center of the Earth, you would indeed weigh 36 million times as much as you do on the surface of the Earth, say, at sea level?
Ann
