Phew, I have A LOT to learn about routing.
@mrdoornbos my tip as I learned in trade school and confirmed by experience:
Use one layer for only horizontal traces, one for only vertical traces. Don't mind placing extra vias. Be almost religious about it. Avoid diagonals as much as possible.
@stefanie I did start with the noisy part on the top of the board, and the parts that behave better on the bottom, so it's a start ;-)
@mrdoornbos one more thing: if you are using KiCad, you can omit the fill areas until the very end. It keeps traces more visible, and speeds up rendering.
Then at the end, just cover every layer with a fill area.
I usually have ground fill on one layer, and fill one layer with an area connected to the supply voltage. On a 4-layer board, I alternate them.
That saves a lot of traces for the supply.