Gravity in The Core of The Planet

 So uhh, yea, earlier on today, I was wondering about what happens to the size of the force of gravity as you get closer to the core. Apparently it is accepted that the force grows stronger as you get closer to the core. But, is it not true that gravitational force is dependant on the size of mass? If that is the case, then how come as you get to the core, the gravitational field increases? Take this for example (Hypothetically, of course): You make an elevator which brings you right down to the core. As you near the bottom, there will be less mass beneath you, therefore the gravitational force pulling from below will be weaker, and the force pulling from above will get stronger (Which means of course, the total force will be weaker). Now, if you were to be right in the very middle of the planet, wouldn't you be sort of being pulled out the way in every direction? This would equal a total force of zero. Even though going by the whole "stronger as you get closer" theory, it should be stronger than it is on the surface. So... is this not true, then? If not, then care to explain why? I have always been fascinated by gravity, I just haven't looked into it as much as this. I also googled for a little while and didn't really find anything about this. So either I never googled very well, or it is just bullshit enough to not have been talked about before. I dunno. But I am very curious.