Would the effect of gravity on me change if I were to dig a very deep hole and stand in it? If so, how would it change? Am I more likely to be pulled downwards, or pulled towards the edges of the hole? If there would be no change, why not?
The other answers provide a first-order approximation, assuming uniform density (though Adam Zalcman's does allude to deviations from linearity).  (Summary: All the mass farther away from the center cancels out, and gravity decreases linearly with depth from 1 g at the surface to zero at the center.)

But in fact, the Earth's core is substantially more dense than the outer layers (mantle and crust), and gravity actually increases a bit as you descend, reaching a maximum at the boundary between the outer core and the lower mantle.  Within the core, it rapidly drops to zero as you approach the center, where the planet's entire mass is exerting a gravitational pull from all directions.

The Wikipedia article on "gravity of Earth" goes into the details, including this graph:



"PREM" in the figure refers to the Preliminary Reference Earth Model.

Larger versions of the graph can be seen here

And there are other, smaller, effects as well.  The Earth's rotation results in a smaller effective gravity near the equator, the equatorial bulge that results from that rotation also has a small effect, and mass concentrations have local effects.
