Does gravity increase as you move under the earth?

Yes, but not for quite the reason you might think. If Earth were the same density throughout, gravity would decrease as you moved toward the center, as more and more of Earth's mass would be above you. However, Earth's density is not uniform. Earth's core is made mostly of iron, which is much denser than the rock that makes up the crust and mantle. So gravity would increase as you got closer to the dense. Once you cross into the core, gravity starts to decrease as described above until you reach the very center, where gravity (at least from Earth) is zero.