Since this was not stated yet, I would just like to give my stance on it.
All fundamental particles can be seen as excitations of fields. This is true for photons, electrons, neutrinos, etc. Do these fields need a medium in which they propagate? Not as far as we can tell. Everything we see and experience are excitations of these fields, a single one of which we call a particle. This is a photon for photon (electromagnetic) field, an electron for the electron field, etc.
These fields have different properties such as their mass and they can interact with each other. If a particle is massless (rest mass if this is not implicit), it has to travel at the speed of light.
The spreading of light is definitely not the waves spreading in the background of photons, but the spreading of excitations in the field. No medium is required for the field to live in if we regard the fields as the fundamental quantities.
For some (massive) fields, the quantized nature is much more apparent from our everyday experience to us than others. But the quantization is a fundamental property applicable to all of them.
How do we reconcile the wave and particle picture? I would claim that the particle picture is a construct of our minds coming from our everyday experience of the world.
Don't get me wrong: quantization is a fundamental property and if you want to call this 'particles', I guess that is a matter of nomenclature.
But we should just think in terms of excitations of quantized fields. Some excitations, e.g. some Fock state, have more 'particle-like' properties, other such as Glauber coherent states for bosonic fields more 'classical wave'-like properties.