A commonly used analogy is to represent the electric circuit with pipes filled with water. The electrical current is modelled by flow of water in the pipes, and the voltage is modelled by the pressure. This is known as the hydraulic analogy.
Anyhow, if you have a pipe filled with water and you suddenly increase the pressure at one end, e.g. by opening a valve, the pressure propagates down the pipe at the speed of sound in water. Note that it's not the water itself that's travelling - it's the pressure wave.
This is what happens in your electrical circuit. When you close the switch electrons flow into the wire where they bump into the electrons already in the wire and push them along. The voltage wave (analogous to the pressure wave) travels along the wire at somewhere between a tenth and a half of the speed of light depending on the type of wire, and reaches the light bulb in a few nanoseconds. That's how the electrons in the bulb filament start moving within a few nanoseconds of you closing the switch.