Set up your studio. Get some students. It is legal to teach and practice at home, too, by the way. Start earning money. Claim it on your taxes. You will receive a friendly letter asking you to please register as selbständig erbwerbend. You go do this. You will get a bill for your AHV costs. Make sure you put about 20% of your income aside for taxes and AHV. You will be fine.

Or, go register before you start teaching.

In any case, Switzerland is a civilized country and no one is going to throw a music teacher in jail for having a few students before registering.
I taught for several years during my studies before even realizing I should register. That was no problem. I just received a bill - rückwirkend .

In terms of insurance. If you rent a studio and keep an instrument you'll need a separate insurance. I had my studio registered at Generali as a Musikschule plus having one concert grand insured. Cost a couple hundred francs per year.