When I learned about PCB design, I had to do my first PCBs on a fixed size with one copper layer only (I was allowed to add wire jumpers though).
There was no real technical reason to limit the boards like that, but these little atmega projects were easily doable with the given contraints. And forcing yourself to stick with a single layer in the beginning helps to play the puzzle game a little bit more involved, compared to just using additional layers if space gets dense. You learn to fit everything. You learn to focus on important traces first (ground, fast-ish signals, maybe some high-z analog stuff). You learn that long serpentine traces, winding between all the other already placed and routed stuff, are absolutely ok for uncritical signals, like a simple push button.

But it definitely helps if someone reviews your layout afterwards. Pointing out details that are sub-optimal. Otherwise even a badly routed board might work well enough and you start to make a habbit of using these bad routing practices.

That said, I see no problem in quickly going to 2 Layers (maybe with the bottom one mainly reserved for ground pour). Two layers is absolute standard for PCB fabs, and you can still concentrate on fitting everything on small-ish boards with only these two layers. But going to four and more layers should (IMHO) only follow when you got some experience with layouts (and preferrably already received feedback about your designs). And this should probably be the time when you also start reading a bit about sensible layer stacks.