Which documents should I have with me when crossing borders?

Carry your passport with SOFA card/stamp, military IDs, and (if possible) PCS orders. Everyone traveling with you should be listed on the orders. Remember that you must always take your passport when crossing borders, even in the Schengen Area, where passports may not be checked.

If you are driving, you must have an International Driving Permit, as your USAREUR license is only valid in Germany. The first step to getting the IDP is with an appointment at the Driver Testing/Licensing office.

Source: https://home.army.mil/stuttgart/index.php/my-garrison/all-services/drivers-testing/driving-germany