A great way to buy a second hand "Fahrrad" is to use internet sites such as ebay-kleinanzeigen or craigslist.de. Both are good to find cheap bikes.
Main advice – choose wisely and be attentive while checking bike's working condition. Do not rush the buy.
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In Aachen people in need can buy a second hand bike rather cheap at a local welfare group:
http://www.wabe-aachen.de/index.php/arbeit-u-soz-integration-mainmenu-50/beschftigungsproj-mainmenu-52/fahrrad-recyling-mainmenu-56