Germany has some coverage of the public WiFi network (the U-Bahn in Berlin, some – but only some – hotels, museums, and cafes, for example, offer free WiFi service), however it’s not as developed as in some other European countries like Estonia or Austria. Even those places that do offer public WiFi, usually limit the usage to 1 hour or 30 minutes. This odd behavior is caused by the ruling of the German civic court that holds public providers responsible for the unlawful activities of customers using their Internet connection, like illegal downloads. Here is a map of the WiFi hot spots in Germany.
To make sure to always stay connected to the Internet, you have a couple options:
Buy a portable WiFi device, like Skyroam Solis or Huawei MiFi. These devices can have multiple phones connected to them (and sometimes double up as charges), they also work in a lot of countries in the world, including Germany.
Buy a prepaid German SIM-card. You would usually have to put in your address when buying a SIM-card: you can use the address of the hotel you’re staying at. There are a ton of great value prepaid sim packages out there, even grocery stores like Aldi and Lidl sell these SIM-cards, so feel free to shop around. The operators usually have similar data plans. O2, for example, offer the following flat internet data packages:
Internet-Flat data packages lasts 30 days, UMTS/HSDPA will be used up to a specified data-limit, after reaching that limit you are downgraded to GPRS/EDGE. You can buy the SIM-card at any O2 store in Germany.