Schools in Stuttgart

Hi and welcome to the forum.
The German school system has a four-years primary school (for kids 6 - 10 years old), followed by either 5 years "Hauptschule" (for those who want to become manual labourers), 6 years "Realschule" (for more office-based professions) or 8 or 9 (depending on the school) years "Gymnasium" for those who want to enter university.
You cannot choose the school, but have to enroll in the one nearest to your residence. But there are no big differences in schooling quality between them.
Teaching is entirely in German (except language classes, of course) and the public schools are free of charge. Kids generally learn German quickly, but you may have to accept them losing a school year in the process.
International schools are quite expensive and so inconveniently located that you'd have to drive the kids there every day if you don't live near one (which is unlikely). There are also mixed reports about their quality.