A battery generates current according to the resistances within a circuit(V=IR).The charged particles enter and exit the source with the same magnitude, however, since we have said that the current is the same everywhere in a series circuit, what happens in the time interval in which the first particles leave the circuit and enter, thus the current is calculated. Shouldn't current be same during this time, in all circuits regardless of resistance?
Interesting question! Actually, not all of the particles which travel around originated in the battery/current source. As soon as the potential difference is established, particles (electrons, holes, ...) from within the wire connecting the pols of the battery start traveling around. After a negligibly short amount of time, this current becomes relatively homogenious in the wire.
Electrons move around in the wire like crazy, with very high thermal velocities. When current is running, their speeds are just slightly biased towards the direction opposite of the current flow. This effective velocity is just a few centimeters per second.
As Samuel Bosch said, the charged particles don't just "enter" and "exit" from the battery. The wire was already full of charged particles (electrons) which were just moving around at random, and therefore not producing an "electric current," before the battery was connected.
You can think of the complete circuit as being like a pipe full of water, where the battery acts like a pump. If you pump a drop of water into one end of the pipe, it pushes all the water along the pipe, and a different drop of water comes out of the other end, and goes back in to the pump. But all the water in the pipe is the same, so you can't distinguish between the different drops of water.
When you switch on the electrical circuit, the effect of the battery "pushing electrons into one end of the wire" propagates along the wire at (approximately) the speed of light, even though the individual electrons are drifting along the wire very much slower than that - usually only at a few millimeters per second.
You can picture this as something like a Mexican Wave in a sports stadium. Each individual person raises and lowers their arms quite slowly, but the next person along the line starts moving their arms long before the previous person as finished moving. So the "wave" can travel round the stadium much faster than the speed that each person's arms are moving.
Beginners often get the wrong idea that somehow "electrons move along the wire at the speed of light" which is completely false. In your house electricity supply, nothing physically "travels from the power station to your house" when you switch on an electrical device. The wires from the power station to your house might be many kilometers long, but each electron in those wires only oscillates backwards and forwards a fraction of a millimeter, 50 or 60 times a second. Each electron pushes on the electrons that are near it, and the disturbance travels along the wire very fast, even though the electrons are moving very slowly.
Final note: all these analogies and explanations are really over-simplifications, but they are useful to get you started. In real life, there are no electrical circuits which contain only resistance. For example a complete circuit consists of a loop of the wire and the other components, which is exactly the same as a coil of wire with only one turn, and you probably know that a coil of wire acts as an electromagnet!
To fully understand "what happens" at the instant when you switch on the circuit, you need to consider all the details similar to that one example, but that is way beyond a first course in electricity. When you are just beginning a subject, you have to just "roll with it" and accept that some things are "true" in the sense that they are good approximations to what is really happening, but they don't provide a full explanation of what is going on.