In this part of Switzerland the phrases 'good gymnastics school' and 'pretty casual' cannot be used in the same sentence.

In your region it might be different.
The National Orgnisation for gymnastics and other forms of 'sport in the hall' is 'STV', in the French-speaking part of Switzerland 'FSG'. So if you look down the list
here, to some group with these initials, they might be able to help you further. Around here there is 'Muki or Faki Turnen' (Gymnastics for Mother/Father with Child) and the next step is 'Kinderturnen' where the child goes to the group alone. This is all pretty casual.
After that it becomes slightly (understatement of the year) more serious.
Perhaps someone from your area knows more.