So the wave moves like. a wave, it moves up and down, up and down. But how do photons move? Do they follow the same path or do they just go straight forward without oscillating?
The question is a bit tricky. Actually, a photon is the electromagnetic wave. Photons are quanta of the field. You can imagine a photon as the fact that there is an oscillation in the field.
However, if you are in an approximation that allows you to treat photons as particles, then I would say that the don't oscillate, they just move on at light speed.
I know, the answer is not complete, I should explain better how photons can be defined, but perhaps someone can do it better than me.
According to the QM, a photon leaves an atom and reaches after some time another atom. The QM makes no statement how the photon bridges the gap and therefore QM can not explain Doppler effect and interference. A variant of QM, the "De Broglie-Bohm theory", tried to describe the path of a photon, but is considered to be refuted.