Moving to Stuttgart

You should look through the forum because the subject of International schools have been discussed at length. There is just one International School in the Stuttgart area which indeed instructs in English. It is in Degerloch which is not far from Vaihingen. They have another site in Sindelfingen which is about 20 KM further south but also reachable by S-Bahn. It is supposed to be a decent school but I am listing the prices shown on their website (link also below),
Early Years 3 & Early Years 4     €11,650
Kindergarten      €14,930
Grades 1 - 5      €16,180
Grades 6 - 8     €17,320
Grades 9 & 10     €17,840
Grades 11 & 12     €19,090
In addition there is  an annual Application Fee of €750 per student and a one-time Admissions Fee of €7,500.

Obviously it is not cheap! but almost no schools of this sort in Germany are, especially considering public schools are equally as good and free. If one is really well paid and only staying for a year then this might be the best option for you. If staying longer, any child would likely learn German anyway and do so better by being in a public school.

The Vaihingen area, and nearby areas like Degerloch and Möhingen, are fairly quiet. Since they are south of the center rather than in it, one might be lucky and find an apartment or row house with some garden behind which is very unusually in the city center. Standalone houses to rent are very uncommon and likely to be extremely expensive. But there are virtually no really bad neighborhoods in Stuttgart. One might prefer not to be by a busy street but just one block farther might be fine. So one is usually going to look for accommodation that is easy to reach from their place of work and/or to school. But if in a public school one just goes to the closest one that fits.

Housing is very tight in Stuttgart, one will not always have an ideal option to be right near their work place or school. So a real consideration can be access to public transportation. It is pretty good in Stuttgart but of course being within walking distance or a single bus or U-Bahn ride is different than needing to change multiple times to commute.

And the subject of finding English speaking jobs have been covered numerous times. Unless one is in a very niche field like IT or certain high level positions working for a large international company, then the opportunities are very limited. English is fine in addition but not as an alternative to German. Well over 90% of jobs will require decent German. It is simply wishful-thinking that English will be enough since it is taught as a second language here.

https://www.issev.de/

There is another school with teaching in English in the Stuttgart area: SIS (Swiss International School) at Fellbach - it is slightly cheaper than the abovementioned and located on the exact opposite side of Stuttgart as Vaihingen is.
But I support Tom's statement above: Unless you plan to stay only for a year or two, there is no better way to assist your kid's arrival, integration and wellbeing here than sending him/her to a normal German neighbourhood school, even if that means losing a year academically due to catching up with the language!

Thank you very much for the feedback. I will look at other posts / discussions. Just one further question, if I may, in regard to schools. I do not mind and will be happy for my son to go to a public school as we will be there for a while, and the amounts mentioned will be difficult for us to cover. But how will his studies be organised in the public school if he does not speak German? We do plan to start learning German before we arrive, and will continue improving our German after we arrive, but I am not sure the limited level of German will be enough to be able to continue his studies. Can you advise on this please?

And for jobs, I am looking for international organisations. At the moment I am working for a British Organisation and have a very high / leadership position. So I suppose it will be possible to find a job at an international organisation until I am able to improve my German.

Depending on their age and daily exposure, kids learn a language pretty fast. Read about "Förderklasse", or special support courses to bring foreign kids up to the level (not only in the language) needed to enter regular classes. Many public schools offer "Förderklasse". Of course, depending on his age, ability and dedication, he will lose several months to a year academically for this, before he is able to resume normal lessons (in German). But I believe this time is not "lost".

Regarding your job: Stuttgart is NOT a centre of international organisations, where you may find one of the (few) niche jobs you are dreaming of. (You'd have to move to Berlin or Frankfurt for that!) The only jobs I am aware of that are available without good German skills are manual (supermarket shelf stockers, harvest helpers, etc.) and specialised technical jobs that are difficult to fill locally (e.g. in certain IT areas). For all others, especially those that involve at least some interaction with customers, suppliers, partners, colleagues or superiors/subordinates, German is expected and essential to perform one's duties. This is because, even though most Germans speak and understand some English, they are not as comfortable and able to express themselves in it as in their mother tongue (which is always the lingua franca in the office, even in the few officially English speaking organisations). It is the same in daily life, where you will be able to survive in English, but face difficulties finding close German friends or building a meaningful social life.
There is no alternative to learning German when moving here. I recommend you invest your first few months in intensive language classes, rather than searching for a job that might not exist!