Metals consist of small crystals; within each crystal exists the "free electrons" which are shared by all of the atoms in the crystal lattice. The number of free electrons per atom depends upon the details of the atoms, but is most often 1 or 2. The free electrons are visualized as "the electron sea" in the Drude model, devised ~1900, and is semi-classical. Introductory condensed matter texts often start with this model. The situation is slightly more complicated with alloys, but the same ideas hold.
In the electron sea the electrons are electrically shielded from each other by (a) the net positive charges of the ion cores and (b) the uncorrelated motions, essentially random, of the free electrons, which are described using the kinetic theory of gasses.
The crystal boundaries serve to impede the free flow of these electrons from one small crystal to the next, and also serve as scattering sites which continually randomize the motions. The velocities of the free electrons are quite large. When an external electric field is applied, it appears as a net "drift velocity" in the electron sea. This is the current in that piece of metal.
When you bring two clean pieces of metal together all of the above is still true, but there is an additional restriction: each crystal has an effective "crystal voltage" on its interior, and for crystals of the same type it should be the same. But when different metals are joined, the difference in the crystal voltage causes a voltage drop when going in one direction, and an increase in the other. This voltage difference is known as the Seebeck effect, discovered in 1821. Since the internal voltages change slightly with temperature, it is possible to measure temperature change electrically; this is the physical basis for the thermocouple.
So adding additional metal increases the total resistance of the circuit, depending upon the resistivity of the additional metal, its dimensions, and other properties.
The current is the net flow of electrons; each individual electron barely moves, but the effects are passed down the line. With alternating currents for every move forward, there is a corresponding move backward -- hence no net motion from the electron drift at all.
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Let me reply with the hydraulic analogy, i.e. with replacing electrical current by water flow.
Open the water tap in your kitchen. Then water comes out instantaneously, although the waterworks feeding the water pipes might be many miles away from your house.
Of course, this is not surprising. Before you opened the water tap, the water was already present in the pipes all the way from the waterworks to the water tap in your kitchen. It is the pressure, not the water, which propagates so fast (theoretically with the speed of sound) through the pipes.
This water scenario above is very much analogous to the electrical scenario.
When you switch on the light in your room, the electrical current through your lamp begins to flow instantaneously, although the electrical power station might be hundreds of miles away from your house.
This is not surprising here, too. The electrons were already present in the wires all the way from from the electrical power station to the switch and the lamp in your house. It is the voltage in the wires and the electromagnetic field around the wires, not the electrons, which propagate so fast (theoretically with the speed of light).