Whenever we switch on a bulb......it takes almost no time to glow up.....But we know that the atoms of a solid are tightly packed and there is a very little space between them. So how the electrons travel through them irrespective of so much blockages in the conductor???
The electrons themselves don't move all that fast. The wave energy is the part that moves quickly.
Picture it this way.
You have 500 meters of pipe, with a small hole at the other end. The pipe is full of water and you increase the pressure at your end. Water will flow out the other end immediately. This is the electrical energy (pressure) and the copper(water).
Now add some dye to the water and note how long it takes for the coloured water to come out the hole. The dye represents the new electrons, and they take noticeable time to move through the system. Pressure moves immediately.
(ok, in this case pressure moves at the speed of sound in water)
In fact, electron's speed is not so fast that light bulb glows up immediately. It is the electromagnetic field which travels in the circuit at near the speed of light that is resposible for it.
After turn on the light, electron only acquires a little speed in addition its thermal speed. The thermal speed of electron can be estimated by $mv^2/2\approx k_BT/2$, where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature and nearly 300 K. So the thermal speed is about 67.4 km/s.
In the electric field inside the circuit, the distance the electron travels freely before hitting the nucleus is estimated to about 300 nm in copper. If the voltage is 110 V, the speed increases by 86 m/s. In fact, the increase is the upper bound, because I assume the voltage change in 300 nm is 110 V, which is not true. Voltage changes over a much longer distance.
So you see, the increase in speed of electron is only about 86 m/s. The draft speed of electron is not really hight (compared to thermal speed).
Electrons can sneak pass all the atoms because of their wave function. They behave like waves not like particles. In short, because of quantum mechanics. In a periodic assembly of atoms like metallic solid they should not feel any resistance when moving through but because it is not perfectly periodic they feel aperiodic potential and this is why they scatter. They scatter on vibrations of atoms! And, they move really fast but they have no definite direction. When you apply electric field to that metal, signal travels through the wire and all electrons practicly at once start moving in one direction, but slowly. This is known as drift speed. While they still move chaotically and with great speed, they slowly drift in one definite direction and it is this speed that that adds up to the original and still existing chaotic motion that is related to current and our harvesting energy of moving electrons.
You might be asking how metal is such an efficient conductor.
Some of the electrons move freely as a fluid. They are not locked in place around the atoms, and don't need "room" in the classical sense.
Here is a wikipedia page going over the real details.
On a scale much larger than the inter atomic distance a solid can be viewed as an aggregate of a negatively charged plasma of the free electron gas and a positively charged background of atomic cores. The background is the rather stiff and massive background of atomic nuclei and core electrons which we will consider to be infinitely massive and fixed in space. The negatively charged plasma is formed by the valence electrons of the free electron model that are uniformly distributed over the interior of the solid