First, forget everything you've ever heard about photons. It's misleading.
Consider the view that light is an electromagnetic wave. This view describes 99.9% of all properties you will ever encounter with light, including diffraction, refraction, and reflection. It is only when you look into how light interacts with matter that the "particle" properties come into play.
It turns out that EM waves only interacts with matter in specific amounts of energy at a time, an amount that depends on the frequency of the wave. The higher the frequency the higher the energy that the wave interacts with. This "packet" of energy is what we call a photon. This is different than the situation in a classical wave, in which energy is transferred continuously, not in packets.
So no, the photons are not traveling in waves. They are merely the way an EM wave interacts with matter. It is the EM wave itself that is doing all the moving and traveling and oscillating.