I would say classical wave (equations) are about propagation. One can distinguish between propagation in time, and propagation along a ray in space. The group velocity is typically closer associated to propagation in time, while the phase velocity is closer associated to propagation along a ray in space.

For quantum interference effects, often the phase velocity is the more important aspect. In order to allow you to focus on that aspect, one considers the quasi-monochromatic case.
The best way to visualize a single photon for that case is as a solution to the time-harmonic Maxwell equations for the given frequency (i.e. energy) of the photon.
The group velocity (for wave packets) is also well defined in the quasi-monochromatic case (given the dispersion relation), and it should be possible to work-out a good way to visualize it too. (I would have to google, or think about it.)

However, if you want to "break-away" from the quasi-monochromatic case, already understanding or visualizing classical wave phenomena is not easy, but certainly possible. And I doubt it will get easier for the quantum case. Also note that lasers are well described by the quasi-monochromatic case. So instead of trying to get more general, it would probably make more sense to try to build a separate mental model for thermal light sources. I guess an incoherent sum of quasi-monochromatic waves is not bad. Just understand that this is not the most general case (i.e. it is still a quasi-stationary case), but it seems to be a good model for a thermal light source.

I think if you want to build intuition for physical waves, it's best to start with the simple cases -- with waves on a rope, waves on drum head, waves in the ocean. Using your arms to make a wave. These are transverse waves. There are also longitudinal waves like sound waves or certain kinds of waves from earthquakes.

The main intuitive thing to grasp with these waves is the dichotomy between the harmonic motion of things (stuff going up and down or back and forth) versus the actual propagation of the wave (the wave going from source to destination).

There's many excellent illustrations of how these waves work to help build intuition.

Due to the wave nature of motion, you will see effects that are typical of waves. These are refraction, diffraction and interference.

After you understand these kinds of simple waves and their typical behaviors. You will see that other "things" (using this in the loosest possible terms) also exhibit similar behaviors which is why we ascribe a wave nature to them. For example, electromagnetic waves and QM waves. In these cases, however, the "what" (the ontology) becomes more complicated.

But don't let the complications from more advanced topics prevent you from building a physical intuition.

If you care more about the mathematical treatment of waves, then you would want to study the mathematics of oscillatory motion, trigonometric functions (sin, cos, tan), and fourier analysis.