In electromagnetic radiation, how do electrons actually "move"? I've always pictured EM radiation as a wave, in common drawings of radiation you would see it as a wave beam and that had clouded my understanding recently.
Illustration on the simplest level:

Which obviously would not make sense (to me), as electrons would collide more likely moving as such.
For example, in a 10 meter (kHz) radio wavelength, do particles electrons move forward and back ten meters? If so, in which direction, and if in one why not any others?
What does wavelength actually have to do with its movement? Does it change the polarity, make it go in reverse or does it continue the same as others, higher frequency just means "more energy"?
 A: There is no particle displacement in electromagnetic radiation 
(Or any other wave motion for that matter).
You can understand this as following:
A time - varying electric field produces magnetic field, similarly a time - varying magnetic field also produces an electric field.
In EM radiation we have both fields continuously inducing each other. This couple of electric and magnetic fields is what travelling in space.
Hope this helps.
A: In EM radiation, there are no electrons involved (well, there usually are electrons moving around in the antenna that produces the radiation, but not in the radiation itself).
So... what do these "10 meters" refer to? That's the so-called wavelength. EM radiation travels in waves, but now what does that mean? Let's first go to another type of waves: Water waves.
If you look at a bunch of waves and measure the distance of their crests to each other, you get the wavelength: The picture below shows a snapshot of a wave, and $\lambda$ denotes the wavelength. 

If, on the other hand, you would stay in one place and count how often at that specific point the water rises up and down in one complete cycle and if you count the cycles per seconcd, that would give you the wave frequency. 
Now, in electromagnetic radiation, what is moving up and down is not actual matter. It is just the strength of the electric and magnetic field at a particular point. Imagine you had some fancy measurement device that would tell you the strength of the electric field. Then if you'd keep it at one point in space, it would oscillate between a maximum and a minimum with a certain frequency. For radio waves, that's usually around $100 MHz$, i.e. 100 Million cycles per second. 
If, on the other hand, you could record a snapshot in time of your electric field and compare how far apart two maxima are, you would obtain the wavelength. 
So, what's "moving" around are the electric and magnetic fields, not actual charges. Thus, drawings of radio waves as beams of waves are accurate pictures of what's going on, unless you go to very very low intensity radio waves where you have to start thinking about the quantum nature of EM radiation...
A: Typically electromagnetic radiation starts with movement of an electron, a charged particle. Either as a varying current, say in an antenna, or within an atom when an electron drops to a lower energy state and changes shell within the atom.
In either case there is change in movement of the charge, which emits energy as a photon, aka a quantum of electromagnetic radiation or light wave. 
The light wave moves off at the speed of light, and depending on the energy involved, which determines the speed of the electron's movement, the wave will be produced over a longer or shorter period of time. This determines the wavelength and frequency of the wave, with more energetic photons being produced more quickly and so having a shorter wavelength than their less energetic cousins.
So the electron does not have to move far to produce a long wavelength, it just has to move with less energy loss. 
