The city centre itself is not very big and you are able to reach the different attractions there by foot.
Bonn has an excellent bus, night bus, tram and subway system operated by the local Stadtwerke Bonn (http://en.swb-busundbahn....). There are ticket offices and vending machines at major stations, offering single tickets, multiple tickets and both 24-hour and weekly passes. The tickets are valid in local trains, S-Bahn, tram, underground, buses and night buses.
It is also a good city to explore by bike. Rental bikes are available at the train station from Deutsche Bahn at platform 1 or from the Radstation behind the train station (http://caritas.erzbistum-...).
