Since you specify in the title that you're asking about ways to "aquire gravity", I'm going to assume that by "natural gravity" in the question body you refer to a gravitational acceleration substantially closer to Earth's 9.8 m/s2 than Earth's Moon's 1.6 m/s2. (Neither value is any more "natural" than the other; they are just a consequence of the size and mass of an object, in these cases a celestial body.)
In other words, you are seeking a way to increase gravity relative to that on the surface of the Moon, and are asking if going toward the Moon's core will achieve this objective.
Newtonian mechanics tell us that any amount of mass will exert a gravitational force on all mass around it, diminishing with distance. (This could also be analyzed through relativity, but Newtonian mechanics are sufficient here.) Therefore, it might seem logical that, by going closer to the core, you would be going closer to some of the mass of the body, thereby increasing the effect of its gravitational force.
However, by doing so, you also place mass above you, which is going to exert a graviational force upwards, just like Newton's famous apple exerts a gravitational force on the Earth it's falling toward.
The extreme case of this is being a point mass at the exact center of a perfectly uniform sphere, itself not inside any gravitational field; such a point mass would experience equal gravitational force in every direction. In such a case, even though it has mass, it would be weightless. (It wouldn't remain weightless for long; the slightest perturbance would shift its position in some direction, at which point the gravitational force would be ever so slightly greater in one direction than in all others.)
The other extreme case is a point mass at rest on the surface of that same sphere; in that case, the gravitational force experienced by the point mass would be toward nadir ("down", toward the center of the sphere).
As the point mass goes from the surface of the sphere to the center of the sphere or vice versa, the gravitational force apparent to it will gradually change from one extreme to the other, but not beyond either.
Therefore, going deeper into the Moon will not afford a greater apparent gravitational force than being at the surface; in fact, quite the opposite.