How Fast Do Electrons Travel In A Copper Wire ?

When you press on a light switch, you can see a lamp that is far from you, turn on in an instant, so you could guess that the speed of electrons moving through these copper wire that connects the switch to the lamp would be very high, but that is where you've mistaken. 

"Electron Drift" is the term used to describe the speed of an electron moving through a wire or any other conductor.

If the cross-sectional area of a copper wire is equal to 3.00 x 10^-6 m^2 and the current is equal to 10A , the drift speed through that wire would be  2.5 x 10^-4 m/s or about 0.25 millimeter per second. It could vary from 0.1 to 0.4 millimeters per second based on the current and the area ... So if we compare that speed to an average snail which travels around  3-10 millimeters per second, the snail would be much more faster than that single electron.

Why Do Electrons Travel At A Low Speed In Conductors ?

When a single electron travel through a conductor, the path it make isn't a straight line, instead electrons move in a random zig-zag motion, which causes the electron to change direction as they collide with the atoms of the conductor, which make the drift speed of these electrons through a conductor very small.

If Electrons Move So Slowly, How Can A Switch Turn On A Lamp Instantaneously ? 

Think of the copper wire as a pipe completely full of water, if you force one drop of water in one end, that will cause another drop of water to drop down quickly at the other end of the pipe. While that drop of water you forced hadn't traveled the whole pipe it is the same for electrons in conductors, in fact if you want to watch a single electron travel from the switch to the light bulb you would have to wait for an hour. But what really happens when you turn on or off that switch is that you cause a small movement in the great amount of electrons  packed so tightly into the copper wire (the number of electrons in the wire is 8.5x10^28 per cubic meter) and thus the electrons will flow at the other end without having to wait for a single one to travel the whole path.

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References :  Slower Than a Turtle - The Speed of Electricity