Don't be afraid to share a table with people especially at a large picnic table in a beer garden. It's polite to say hallo or Guten tag and then asking if the table is free. Once you sit down and if they get some food it is also customary to say "Gut appetite". You don't have to chat with them but politeness if very important.
When entering a restaurant that is not touristy and not that large it is customary to say a polite hallo or Morgen to the diners. When you walk into small bakeries or butcher shops/stores it is always polite and customary to say hello when entering.
My impression is that the people of Stuttgart are very nice and polite. They may hesitate to speak English; maybe they feel that that their English is not good enough? So if you speak German it will help you. Otherwise I would advise you to speak slowly English. Don’t ask if they speak English, they may say that they don't speak it. Try to make your English easy and slowly and you'll probably find someone that can help you.