Exactly at the center-of-mass of the Earth there is no net gravitational force. Gravity comes from mass, and all the mass of the Earth is evenly distributed around the center-of-mass (by definition), so all the forces cancel out. As you get deeper and deeper, the net gravitational force falls -- linearly if density is constant.
In practice, of course, you can't dig a hole that deep; much of the interior of the planet is flowing molten rock.
If you could magically make a stable, not-ridiculously-hot tunnel through the planet, and you and your antipodean friend each jumped into one end of the tunnel, you'd fall toward the center and meet there.
According to a crude simulation I've worked up, if standard sea level air pressure was magically maintained throughout the entire tunnel, you'd be moving at somewhere around 0.25-0.75 meters per second when you reached the center -- it depends on whether you spread out to catch as much air as possible, or straightened out in a headfirst dive to maximize your speed. In the median case you get to the center moving at 0.5 meters a second after a 48-hour fall! That's slow enough that you and your friend should be able to hug without injury when you meet. My simulation assumes constant density of planet all the way down, which is obviously not right, so you'd probably collide a little bit faster than that, but not greatly so.
If you pumped all the air out you'd be accelerating all the way down, but as you got closer to the center and more and more of the Earth's mass was "behind" you, your rate of acceleration would reduce; just as you got to the center the acceleration would tend toward zero, but of course your accumulated velocity would still be extremely high. It would take ~21 minutes to fall, and you'd reach the center at ~7900 meters/sec -- a figure which is suspiciously close to circular orbit velocity, by the way.
With no air and no obstruction in the tunnel, you'd shoot through the center, and the acceleration you experienced on the way down would be exactly reversed on the way up the other side. When you reached the opposite end of the tunnel, your velocity would be back to zero, and you'd start falling back to the center. Without any drag or friction involved, the end result would be that you'd bounce back and forth from one end of the tunnel to the other forever.
If you lowered yourself carefully down a 6400 kilometer cable to the center and let go, you'd be in free-fall at the center and wouldn't go anywhere.
Here's the Python code for my sim.
# initial altitude is earth's radius
radius = 6371000.0 # meters
alt = radius # meters
# start motionless at T = 0
vel = 0.0 # m/s
t = 0.0
# time step, seconds; I get basically the same result with either 1 sec or 0.1 sec
tstep = 1.0
# air drag parameters; CD and sectional area are
# selected arbitrarily - they can vary enormously
# depending on the skydiver's pose.
rho = 1.225 # kg/m^3
cd = 0.4 #
area = 0.6 # m^2
mass = 80 # kg
# quick switch between the vacuum case and the air drag case
tunnel_has_air = True
# Gravity's the easy part; if we pretend
# Earth is constant density, then g is linearly
# proportional to altitude.
# This would break if altitude goes above Earth's surface,
# but it doesn't in this sim.
gravity_accel = 9.81 * alt / radius
vel -= gravity_accel * tstep
# Compute air drag
drag = cd * area * 0.5 * rho * vel * vel # force
decel = drag / mass
# get the sign right
if vel > 0:
vel -= decel * tstep
vel += decel * tstep
alt += vel * tstep
# Update time
t += tstep
print "t %.1f altitude %.3f km velocity %.2f m/s" % (t,alt/1000.0, vel)
# When we get to within a decimeter of the center, we can stop
if alt < 0.1: