How fast is electricity?
It is not the electrons that are flowing through an electric
wire, it is a charge (energy) transferred from electron to electron
within the wires. But even this motion is much slower than you
think, and depends on different factors, such as transfer medium.
Even lightning traveling through ionized air can only travel, at
most, half the speed of light. (see the related link)
There are several different ways to define the speed of
electricity. In an AC circuit the electrons do not move forward at
all, they just vibrate back and forth in place.
At the same time, the amount of time that it takes between when
you flip a switch and a light turns on is actually much faster than
the speed of light, so that it actually takes longer for the light
of the bulb to reach your eyes than it takes for the light to turn
on after flipping the switch.
The easiest way to imagine this is to picture a mile-long pipe
that is just larger than a marble and is filled from beginning to
end with marbles. When you put another marble in one side, a marble
comes out the other side nearly (but not quite) instantly. This has
nothing to do with the "speed of marbles," just like electrical
current has nothing to do with the speed of light.
Other things such as radio waves, magnetic fields, and so on are
a completely different story.