It's funny you should ask this question. I remember a homework assignment in college in which we were asked a similar question. The question we had was, what would happen if you dug a hole all the way through the earth(from one side to the other, right through the middle) and dropped an object into the hole? The answer is as follows.

We dig the proverbial hole to China. We toss a stone into the hole. The stone is pulled down towards the center of the earth. It falls all the way to the middle of the earth once it reaches the middle, theres no more pull of gravity. But the stone is going pretty fast, and there's nothing at the middle of the earth to make it stop (since we dug all the stuff out). So the stone would keep on moving, going straight on up the other side of the hole, towards China. It would go up a ways, then (due to the pull of gravity) fall back in the other direction, back towards the center. This back and forth oscillation around either side of the middle point of the earth continues for a while. Eventually it stops, because all the while there's air that slows down the motion of the stone. The stone would stop right in the center of the earth.

In the homework problem I mentioned earlier, we were asked to assume that the earth has a uniform density throughout, and that there's no air resistance. (Physicists often make unrealistic assumptions like this.) We then found that the stone would oscillate forever, without stopping. Each time it moved back and forth, it would come all the way up to the surface of the earth, before falling back down again, and each cycle of oscillation would take about an hour and a half. In other words, if you dropped the stone in the hole, an hour and a half later it would come back up to the same spot you dropped it from. Pretty cool, but keep in mind that it wouldnt actually happen like that because the air resistance that we conveniently ignored actually has a significant effect. It would actually happen more like I described in the second paragraph, above.

You might have a little difficulty digging a hole to Earth's center,
since the core is molten, but the idea is intriguing.If you tried
instead to dig a hole to the center of a body that does not have a
molten core, like a small moon, you could actually test your
hypothesis. In either case, molten or not, at the center of a
spherical body of uniform density, the gravitational attraction is
evenly distributed in all directions. The net result is that if you were
to crawl to the center, as you approached the center you would feel
progressively lighter. At the center, you would float. Now, imagine
that you bore a hole straight through the center from one side of
the small moon to the other. If you stood at one mouth of the hole
and dropped a rock into the hole, what do you think would happen?

You would indeed float at the center of the earth (though I dare say it would be hot at the core of the earth!). You might not want to jump all the way from the surface, though, unless you dug your hole all the way to the other side of the planet, because you'd hit the bottom of your hole still falling.

Here's a question for you to try: Pretend the earth is a uniformly dense perfect sphere of matter, and drill a hole straight through the earth(small enough that it doesn't affect the gravitation much). Try using Newton's laws of mechanics and gravitation to relate motion through the hole to the motion of a weight attached to a spring.

The simple answer to your question is that it would be very difficult to dig a hole to the center of the Earth because the immense pressure at depth would cause the sides of your hole to collapse. The interior of the Earth is also very hot so you would get cooked on the way down. Of course that's no fun and doesn't get to the core of your question.

The mass of the Earth exerts a force on everything, pulling objects towards its center which is why we fall until we run into some immovable object between us an the Earth's center (for example, the ground). If there were a hole all the way to the center of the Earth, we would fall, accelerating
until we reached terminal velocity (where the resistance of the air prevents us from going any faster). The problem is, we have momentum when falling so we would go right through the center of the Earth. If everything is being pulled toward the center of the Earth and our hole goes all the way through the Earth, what do you think will happen after we fall past the center of the Earth? Will we shoot out the other side or slow down?

Ah, a very good question. This is often asked in a Physics class.
Because of the way gravity works, the gravitational force that an object feels (say you) due to another object (say the Earth) is proportional to the mass of each object. However, if you are traveling through a tunnel in the Earth, then the gravitational force that you feel is not proportional to the entire mass of the Earth but to the amount of the mass that is contained within a sphere that is centered at the center of mass of the Earth and has a radius equal to the distance you are from the center.

I know this is very wordy. To put it another way: as you travel through your tunnel towards the center of the Earth, the amount of gravitational force you experience decreases from the surface value (your weight, in pounds for example) until at the center there is no gravitational force on you at all. A way to picture this is that if you are at the center of mass of the Earth there is an equal amount of
matter surrounding you in all directions. Why would you be pulled in a particular direction?

So let's say you jump into your tunnel. You start falling freely and accelerate towards the center of the Earth due to the force of gravity. As you fall, your speed keeps increasing but more so at the beginning than as you approach the center because the amount of gravitational force pulling on you is decreasing. When you reach the center, your speed is not increasing at all but you are still moving very fast. So what happens? Do you get stuck in the center?
Nope. You overshoot the center and keep on moving. Now gravity is pulling you in the direction opposite to your motion and thus slows you down. A little at first and then increasingly as you get closer to the surface until finally, at the surface, you pop out and your speed is just around zero. If you want to, you can jump back in and pop out on the other side again.

This of course assumes that you can dig the tunnel and survive the heat during the trip. It also assumes there is nothing opposing your motion such as air resistance. What do you think would be different if there was air in the tunnel to oppose your motion?