becoming a teacher in public schools up to and excluding the university level isn't easy in Switzerland (it's not easy in the US or Canada either, or so I'm told).
In order to teach in a public school, you'll have to obtain a teacher's degree. This involves studying for another 1 1/2 years at a college for education, a couple of internships and a final examination in German.
Unless you already took some psychology and didactics classes during your time at the university, you'll more than likely have to go through the whole curriculum.
I'm not sure what you need to do to teach at a University - as far as I know you don't have to have a teacher's degree for that. Normal "Lehrauftraege" (university teaching positions) are usually only part-time and salaries are very low.
Generally, you don't need a teacher's degree to teach in private schools. But private schools usually have major drawbacks in terms of employment contracts. They're usually at wages-per-hour basis, don't include paid vacations (and you have to take +- 12 weeks per year of unpaid vacation), no bonuses and only very basic pension plans.
It is somewhat similar to the us (California that is). You take a CBEST and take an internship. Overall that can take 2 years.
What surprises me is that (at least in the US) they are complaining about shortages of teachers. Apart from the sub par pay on public schools, they
do not value the education and work experience somebody can bring to the job.
If I look at the Netherlands (where there is also a teacher shortage), they
offer programs for experienced professionals to switch to the teacher
profession that matches their professional background, where they can start out as a teacher (usually starting teaching the lower classes) and follow the necessary courses in parallel depending on their background.
My experience with teachers on a high school level (in California that is) has been somewhat disappointing specifically in being able to teach their subject (engineering related disciplines) on an adequate level. As a result
universities need to pick up the 'slack' in the lack of knowledge. A similar trend seems to take place in the Netherlands when talking to university lecturers.
I am wondering how this all relates to Switzerland.
Well, here it depends on what you teach. You're actually in luck - there's always a shortage of math teachers. Sometimes, schools will let you start teaching without having the degree - you can than take the necessary classes while you're already employed.
My wife's a teacher for English and German. Unfortunately, there's an abundance of language teachers. It was very hard for her to find a job after getting her teacher's degree. She has now been hired but only on a temporary basis for 4 years - after that, they'll decide whether they'll keep her permanently or fire her. Needless to say: most of her colleagues who already went through the four years of "probation" got fired at the the end, because it's quite hard to get rid of a teacher once he or she has a full-time position. Because student numbers constantly fluctuate, they prefer the temporary contracts.
Some of my friends who studied maths/physics at ETHZ/EPFL went into maths teaching at secondary school resp. high school after their degree (the old Diplom, sort of equivalent to Masters level). I'd have to double-check, but I believe that all they had to do, was to follow some didactics courses for a year during which they were already giving some classes (none of them had taken such courses while at uni). I think they were already earning some money as well. This was about 4 years ago and in the Ticino region (where the rules regarding teachers are probably not as tight as in the German-speaking part). I don't think that it matters whether or not you've got a PhD: two of these friends are actually PhDs in maths, resp. physics, are teaching at high school level, and had to go through the same process as the people with only a degree.
I doubt that you can work as a teacher at university level, unless you're employed as a researcher: as far as I know, the position of plain lecturer does not exist in Switzerland. (Well, let me know if you do find out that such a position exists! I'd love to teach, if at uni level!)
Well, there are the so called "Lehrbeauftragte" (sorry, don't know the English word for that). Those are external people that teach some classes at a university. They're usually employed elsewhere and only teach a couple of hours per week. Theirpay is somewhat symbolic
In private schools, it shouldn't be problem, since the school is abilited to decide what curriculum their teacher should have. They are indenpendant for that.
For public schools you should first get a swiss equivalence to your degree.
Once you have your diploma compared to what is required here, you will know if you'd have to adjust / complete your competences or not, and at which degree you could teach.
A good start to get acquainted with the school system here would be IMO to go to schools/university and offer private lessons in maths on their message boards or beeing a substitute teacher. So you could know what to expect.
I definitely would do that, since as a teacher here, you can't quit your job after a couple of months like in other professional fields if it is not what you are looking for, but you have to continue till the end of an entire school year.