In metallic wires
The moving charged particles are simply part of the atoms that make up the battery, wires and bulb. When metal atoms combine to form a wire, the result is a lattice of fixed positive ions and and electrons that are free to wander.
The atoms were neutral, therefore the wire will be neutral as well, as charge is conserved. These electrons do wander, at high speed, randomly whizzing about in very short hops, changing direction often, with different thermal motions (100–1000 kilometre / second).
When these components are connected into a circuit, a steady very small drift velocity (a few millimetres in each second) is added to the large thermal velocities (the movement of the electrons is rather like the movement of molecules in a cloud of gas drifting sedately along). The charged particles that were already in the wires now move consistently, in addition to their random short hops.
The electric current in wires is made up from millions of minute, negatively charged particles called electrons, which drift along the wires. As a result the charge flows around the circuit.
Here are two representations of this movement. Current is always represented by an arrow which points along the wire or other circuit element, so that it is pointing away from the positive terminal of the battery and towards the negative. The current arrows point in the opposite direction to the charge flow because electrons are negative.
We refer to the electrons drifting for a very good reason. The additional motion towards the positive terminal causes them to move only about 1 centimetre in each minute. This is very slow: take a minute out from reading this to move something one centimetre across the desk in front of you. This speed is millions of times slower than their random (thermal) buzzing around, so we have simplified the diagrams, by not showing the buzzing around. We don't suggest you take another minute out to mark out the 600–6000 km they will have travelled, as it is very far from a straight line.
In a metallic conductor there are lots of electrons but they move around the circuit very slowly.
In other cases we can perfectly well get a current made up of the movement of charge carriers that are not electrons. But electrical current is always a movement of charge.
There are, however, other situations in which the current is not carried by electrons. For example, a salt solution will conduct an electric current, and here the moving charged particles which constitute the current are provided by the ions in the solution. Currents in nerve cells are the movement of ions.
In the following nuggets we'll refer to electric currents in terms of a flow of charged particles, as this covers all of the situations. In your own teaching we'd suggest that you don't insist on flows of electrons until you can reasonably demonstrate that this is what is happening in a wire.
Find out when the charged particles start to move
It would appear that when we turn on a switch the electric charge moves immediately in all parts of the circuit and instantly lights a bulb. Even if we connect all the wires available in the laboratory, to make a big circuit, the light bulb still appears to react immediately.
Actually it does take a very short time for the electrons to start moving. The electric field that sets the electrons in motion takes a finite time to pass through the wires. The field propagates (moves) at approximately the speed of light: 300,000 kilometre / second, that is 300 millimetre in a thousand-millionth of a second.
Does this delay matter? An electrical signal would take a mere hundredth of a second to pass from the UK to the USA through cables under the Atlantic Ocean. However, a modern computer will time itself on a signal changing many millions of times a second. With signals that are fractions of a millionth of a second long, even small delays in signals travelling along wires must be taken into account when designing circuit boards.
The bulb in our circuit does not turn on immediately, but the delay is so short that it is only significant in the most extreme applications.