Electrons are considered to be 'dimensionless' because it makes absolutely no sense to measure how large they really are. In quantum mechanics, you just can't measure a distance with arbitrary precision. An electron does fill a volume*, and no two electrons are allowed to fill the same volume...
In short, the electric and magnetic fields are the ones waving. Their 'waving' is kinda coupled; i.e. a waving electric field induces a waving magnetic field (described by Ampere's corrected circuital law) and a waving magnetic field induces a waving electric field (see Faraday's law of...