Earth is a lousy place to dig a tunnel through. It's not possible, so why not pick a planet where it is possible. Pick a planet whose core is cold and solid and strong enough to support its own weight without collapsing the tunnel. First consider a planet with no atmosphere and hypothesize that it is as large and as dense as Earth but made of a material like diamond which is strong enough not to collapse the tunnel. To simplify the math, assume it has constant density, despite the compression toward the center. Also assume that the planet does not rotate.

In that oversimplified case, the strength of gravity inside will be proportional to the distance from the center; outside, the gravity will decrease as the inverse square of distance from the center. An object dropped into the tunnel will start accelerating downward at 1 g, and the acceleration will decrease to zero at the center; it will decelerate toward the opposite side, reaching zero velocity at the top. The speed as it passes the center will be the same as the orbital speed of a satellite at the surface, about 8 km/s. It will continue to oscillate back and forth between the two ends. The period of this oscillation is a bit less than the orbital period of a low earth orbit satellite; about 87 minutes.

Now, if you add an atmosphere, the situation is very different. With one Earth atmosphere pressure at the surface, the pressure will increase dramatically as you fall in. It's about 100 kPa at the surface and 200 kPa at a depth of about 10 km. It increases exponentially, so by the time you reach the center (3180 km) it might be something like 10,000 atmospheres. That's just a wild guess. The formulas break down because, at some point, the atmosphere becomes liquid. As the pressure increases, the air becomes more viscous, so you quickly reach terminal velocity of about 195 km/hr, then you decelerate until your are barely moving. You can expect the air to get hotter as the pressure increases, so you eventually burst into flames and your ashes slowly drift toward the center, where they remain forever.

If your planet is rotating, you better drill your hole from pole to pole so you won't drag against the sides of the tunnel.