What happens when we connect battery to a circuit?
So, my question is, does electrons from battery move through out the entire circuit? If yes, what is velocity of electrons. 


Electron flow is actually pretty slow.

What is very fast is the onset of the flow; it's nearly instantaneous in all the metal.

Think about it like a water hose filled with (standing) water: Even if the water flows slowly through it, turning on the pump will nearly instantly make water come out of the end.

So, no. Current is not when the very same electrons leaving one terminal reach the other. Current is the amount of electrons flowing due to the electric field that the voltage causes.

So, while the speed of electrons is pretty slow, the speed at which a change in electrical field propagates is very high – in fact, it's the speed of light.