I'm always asked these sorts of things. Go figure. I suppose it comes with the job, like the Air Force general based in New Mexico who continually deals with Area 51 speculation.
If you dug a hole through the center of the Earth, and jumped in, would you stay at the center because of gravity?
This experimentalist's answer:
No, because (a) the engineering barriers to digging said hole are insurmountable, and (b) if you weren't crushed, you'd be asphyxiated or, more likely, burnt. Look up the temperature of Earth's core.
The theorist's answer:
Assume the hole is wide enough that there are no forces other than the gravitational interaction between you and the Earth. The gravitational field INSIDE the Earth is zero at earth's center, always points toward Earth's center, and gets bigger as a linear function of distance from the center. (The mathematics here are the same as when using Gauss's Law to determine the electric field inside a sphere of uniform charge density. The 1/r2 dependence only occurs outside the sphere.)
By definition, when an object experiences a linear restoring force, its motion is simple harmonic. Thus, you'd oscillate about the center of the Earth like an object attached to a spring. If you jump in from Earth's surface, then, you'd speed up until you passed earth's center, after which you'd slow down, reaching the surface on the other side of the earth before you repeated the process ad infinitim.