Free parking in the old town is hard to find. If you don't mind walking a bit you may be lucky in the streets further out.
If you want to leave your car in a spot close to the centre, though, there are two options (both with meters):
1. the garage underneath Deinzer Platz (off Grabenstraße)
2. open-air parking in Prümer Wall and Himmeroder Wall.
Buses connect Rheinbach's train station with the surrounding villages. Unfortunately the bus connections are not too great and mainly directed to schoolkids and commuters. Outside their rush hours getting around can be time-consuming.
There is also a bus line to Bonn which takes much longer than the trains but runs at hours when the train doesn't. Late in the evening or during weekends you might need it.
Within the town there are hardly any bus lines at all - well, the town is small and mostly walkable. One line goes up to Freizeitpark and the MonteMare swimming pool.
The Voreifelbahn connects Rheinbach with Bonn, where you have connections to the long-distance trains along the Rhine valley, and Euskirchen. The local trains to Bonn run every 30 minutes during the day, even every 15 minutes during rush hour, and hourly in the evening.