Stephanie, from BB's link, perhaps the FBA (Fachspezifische berufsunabhängige Ausbildung) might be the way to go - or at least a good start.
This is the less-intensive of the animal care courses, and it would allow you to work commercially in animal care (for fewer than 19 animals). This is not enough to work as a vet assistant, nor to
run a Tierheim, but you'd be able to build experience in the field which might perhaps pique your interest in further possibilities.
The FBA is a shorter course and includes an internship (unpaid I believe) in a Tierheim/kennel situation. This would allow you to set up your own pet care business or to work in (but not manage) a Tierheim/kennel.
The Tierpfleger diploma is ca. 3 years - this allows you to manage a shelter or kennel. It is a more intensive course of study.
Working in a vet practice usually requires a TPA, a 3 year degree/apprenticeship. This is perhaps akin to what in some countries would be a veterinary nurse.
http://www.gstsvs.ch/de/Ausbildung.htm
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Another possibility for further education that would probably not lead to job qualifications but would likely give you a great deal of personal satisfaction - and perhaps help pinpoint future career goals - is to pursue independent courses/seminars on topics of interest.
For instance, when it comes to dogs Certodog (
www.certodog.ch) is a well thought of training institution - and one of the institutions certified to train SKN trainers if that is a goal.
A foundation I very much respect is Animal Learn in Germany (
http://www.animal-learn.de/index.php/seminare.html).
You could also look at distance courses from the UK or US if studying in English is a preference; you can study to degree level or take independent seminars. I can get you recommendations if you are interested.
BUT these would largely be for personal satisfaction - to work in CH you'll most likely need the CH-specific credentials.
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If you go the FBA route, may I ask that you post a few updates? I've been toying with the idea of doing the course - strictly personal interest - but I fear my written German would not be up to snuff. I'd be interested to hear what others, especially expats who did not go through the CH education sytem, think of the course.
Wishing you all the best!