The Earth is essentially a spherical mass. Your weight is determined by how much mass there is in a spherical volume centered on the Earth's core, and extending in radius to where you currently are located. All of the mass outside this volume does not affect you because the forces cancel. So, as you fall to the center of the Earth, the force of gravity weakens until, right at the geometric center, it vanishes. A caveat is that there will still be the gravitational forces by the Sun and Moon acting on you at even this point. Also, the Earth is not perfectly round, so there would be a residual Earth, tidal force acting on you as well.