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Finding a place to live is not easy in Stuttgart: Rents are high and demand outstrips supply.
The better places are advertised (and quickly rented out) right after the previous tenant submits his termination. Since notice periods are 3 months or longer (by law), this means you have to search (and commit) long before you want to move in. Furthermore, German landlords are often picky, and definitely want to meet you in person before offering you a contract. Thus you need to arrange for a temporary place (a friend's house, or a hotel/B&B/hostel) for 2 - 3 months, come here and then start your accommodation search. You should also ask a German speaking friend to check the rental contract and negotiate with the landlord (if needed).
To find places to rent, you can check classified ads sites like quoka.de, kleinanzeigen.ebay.de, wg-gesucht.de, etc. Ads are in German language, of course.
Budget €350-450/month for a sublet room or €400-600/month for a small apartment (studio). Utilities ("Nebenkosten", like heating, electricity, waste disposal, etc.) go on top of that and cost another €100/month or so.
Good luck - you need that, too!

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