What should I look out for when choosing a grammar school?

You should pay particular attention to this:

Some schools offer grammar school courses for children/young people AND adults, others specialize in one of the two target groups. When choosing a school, remember that it is also important for learning success that your daughter/son or you can feel part of the class.
Some schools only offer short-term grammar schools. Check which school offers a suitable transfer for your son or daughter.
When choosing a school, check whether it actually offers the Swiss Matura or possibly an international version (international Maturas are not necessarily considered equivalent by Swiss universities).
If you want unrestricted access to universities abroad, make sure that you choose an offer that leads to a federal or federally recognized cantonal baccalaureate. The combination of vocational baccalaureate and passerelle certificate is not equivalent to the baccalaureate and only allows access to universities abroad where the passerelle certificate is recognized.