Driving in Germany may be very different than in your home country. It is highly advisable that you read about some of the differences at Expatica and How to Germany
In particular, you will need to make sure you have a driver's license and your car is properly registered.
Environmental Stickers
You currently need a green, 'Umwelt' sticker on your car to drive in Bonn center.
Parking in Bonn & Bad Godesberg
Be very careful parking on the streets as many places may not be legal - even if you see many other cars there, and the parking police are quite efficient. You should only park in marked places and ensure that it is not limited to residents (if you see signs that say 'Bewohner', find another spot!!) Sometimes you will need to go to a nearby machine, pay and print a ticket to leave in your car to show how long you have paid for. Other times, you need to put in a little clock to show what time you arrived, and make sure you don't stay over the designated time. You can buy these little time clocks in gas stations and many other shops. They are usually blue with a rotating wheel with the times.
The safest place to park is in a garage.
Bonn: Marktgarage and Munsterplatz Garages are the closest to the center, but Opergarage, Friedensplatz Garage and Stifts Garage are also convenient to the pedestrian area. Signs indicate how many parking spaces are available. Just take a ticket and enter. Pay at the "Kasse" machine before you drive out.
Bad Godesberg: There is underground parking at the Fronhofer Galeria and underground parking under the Rewe Center in the "Altstadtcenter". There are also garages next to and across from Kinopolis.
