How far do electrons actually move along a conductor under an alternating current?

The electric field travels through wires at practically the speed of light. However, the velocity of individual electrons, known as their drift velocity, is VERY small, on the order of 0.04 mm/s. For more information on this, and a drift speed calculator, see http://hyperphysics.phy-astr.gsu.edu/hbase/electric/miccur.html

Conduction electrons move at very high speeds of millions of meters per second. These speeds cancel out in the absence of a field. When an electric field is applied electron drift results with a very small velocity $v=\mu E$. $\mu$ is the mobility. Typical values are of the order of micrometers per second. To calculate the distance traveled due to an alternating field you need to specify $E = V/l$ where $l$ is the length of the conductor. Note that alternating voltages are given in rms values and that the peak value is $\sqrt{2}$ times higher. Finally the maximum displacement is $d_{max} = \frac{1}{2\pi f}\sqrt{2}\mu V/l$. $f=50 Hz$. For Cu the mobility is about 6 $10^{7}$ Siemens/m.