If it was possible to dig a hole that went from one side of the Earth to the other... ...And you jumped in.
What would happen when you got to the middle of the Earth? Would you gradually slow down, until you got to the middle and once you were in middle would every direction feel like it was up? 
 A: 
And you jumped in.

No, I'd refuse, You should do that Yourself. :=(
Do You want to "solve" this with or without friction by air? 
Without friction, You would fall and reach maximum speed in the center of earth, 
going on until You reach the antipods, where You would stand still for a fraction of a second, 
then You would go down again. Very boring indeed and You should take some sandwiches 
and some drink with You. 
With friction You wold accelerate up to a speed around 300 km/h, then fall down 
with this constant speed, slowing down when approaching the center, and go on some 
distance, fall down to center again and do that oscillation until all Your potential/kinetic  energy is consumed by friction. 
Because about a dozen of such holes were drilled here in physics.SE alone, You might find some company at the center of earth. 
A: I seem to remember Dr Karl on the Triple J radio station (Karl Kruszelnicki) explaining this one. In a zero friction environment, you would oscillate back and forth. It would take about 40 minutes to travel from one side of the earth to the other. An interesting side note was that if you were to dig the hole from any point in the Earth to any other point, it would still take the same amount of time to travel from one side to the other.
A: I'm not a Physics major but it seems to me that at the center of this planet you would have both little to no air and no weight. All direction would be equally down. If how ever you managed to git to one side then the gravity of that side would be half the gravity of the hole world minis the other half of the gravity that is reduced by the distance its source is from you. Gravity not being a constant but being reduced as the distance it is from its source increases. You would also feel the effect of centrifugal force that you could not tell was not gravity. There would be more air here as well but it might not be thick enough to breath. I think that you would feel that you were falling up after clearing the outer skin of the world. The effect of passing the center of gravity for the skin of the world. 
