How do electromagnetic waves travel?
I have a nice picture to show a perturbation of the Electric Field, produced by an accelerated charge, propagating outwards:
I got this amazing picture from my old Physics book (Ohanian Physics):
You can see a charge and the field lines.
The charge is accelerated between P and P' creating a perturbation in the lines that propagates outwards.
Below you have a kind of movie sequence of the propagation of the perturbation.
It also exists the corresponding Magnetic Field perturbation not showed here.