Would gravity become greater, lesser, or stay the same the closer to the center you got?
Gravity's a function of mass and distance. Less gravitational pull when you're on a mountaintop than when you're in the Mariana Trench, due to your distance from the bulk of the earth's mass.
Falling through a hole in the earth, your rate of acceleration would increase until you get somewhere around the core/mantle boundary. Then the rate of acceleration would be reduced, due to the mass of the mantle behind you cancelling out much of the gravitational pull of the core in front of you.
In a frictionless system, you'd cease your now-upwards journey and begin descent at the exact same height that you left at.
Real world problems: friction, intense heat, pressure, the earth being non-homogenous. Last one is an issue as it means you'll likely have sufficiently unbalanced gravitational forces from the sides that your magic capsule of frictionlessness will be slammed into one side of the hole or another as you go down. Bar service might be a bit iffy.