Actually, current moves quite slowly... It's
electromagnetic radiation that moves at the speed of light (radio waves, light waves, microwaves, etc).
Electricity is very analagous to water in a hose. Only difference is wires are not empty like a hose would be- the metals inside are chock full o' electrons (water) already. When electricity is applied, electrons sitting there are just pushed through the wire (as water would be if a hose were already filled), giving the instant response.
This site has some good info on how fast current actually moves.