So, what would happen to gravitational pull if you travelled below that the Earth's surface towards the center? Would your weight in Newtons increase, along with the sensation of getting heavier?
No, quite the opposite. As you go down below the Earth's surface, in a mine shaft for example, the force of gravity lessens. Weight and gravitational pull continue to decrease as you get closer to the center of the Earth.
If you're right in the center, if that were possible, and you've got the Earth surrounding you, then you're being pulled equally in all directions and the net effect is that they cancel out. There's no gravitational pull and you'd be weightless.