So you got thrown in a bottomless pit......

[Perd Hapley]

Lots of physicsy stuff about this very question:

http://www.askamathematician.com/2012/08/q-if-you-could-drill-a-tunnel-through-the-whole-planet-and-then-jumped-down-this-tunnel-how-would-you-fall/

"...by the time you’ve fallen about 20 miles you’re really on fire a lot."   

[Doggy Daddy]

https://www.youtube.com/watch?v=eMUd42n0jXc

[MechAg94]

Wouldn't you at some point have gobs more air pushing down on top of you than you're used to?

That is one of the other reality issues.  I think the Earth averages something like 7900 miles in diameter.  That radius is many times the height or thickness of the atmosphere so yeah, the pressure would be up there.

[230RN]

REFs:
https://www.youtube.com/watch?feature=player_detailpage&v=StHMKdvuHN0

A farmer noticed that his chickens were sick, and called in a biologist, a chemist, and a physicist to help diagnose the problem. The biologist observed the chickens, concluding, "I can tell you there's something wrong with your chickens, but I don't know what's causing it." The chemist took fluid samples from the chickens back to his lab, and returned saying, "I can tell you what's infecting your chickens, but I don't know how they got it." Meanwhile, the physicist had been sitting on the floor, scribbling maddly on several notebooks worth of paper. Suddenly, he jumped up, exclaiming, "I have the answer, but it only works for spherical chickens in a vacuum."