There are numerous museums in Bonn worth visiting. Five of these are conveniently clustered in the "Museumsmeile" area, along the tram line between Bonn and Bad Godesberg:
Haus der Geschichte (German National Museum of Contemporary History), free admission
Bundeskunsthalle (Art and Exhibition Hall of the Federal Republic of Germany)
Kunstmuseum Bonn (Bonn Museum of Modern Art)
Zoologisches Museum Alexander König (Zoological Research Museum Alexander Koenig)
Deutsches Museum Bonn (Museum of science and technology, offshoot of the famous Deutsche Museum in Munich)
And around the central area of the city:
Arithmeum (Museum of mechanical calculating machines and computers), close to the main university building
Rheinisches Landesmuseum Bonn (History and art of the Rhineland), close the the main station
Frauenmuseum (Women's Museum)
Beethoven-Haus (Beethoven's birthplace & Museum)
The Drachenfels (Dragon's Rock) is a hill located in the south-east of Bonn, on the eastern bank of the Rhine near the city of Königswinter (accessible by tram from Bonn).
The ancient castle ruins on its summit and the neogothic castle somewhat below are highly popular destinations for excursions in the Bonn region. The Drachenfels can also be accessed by a rack railway.
The legend tells that this is the place where Siegfried, the hero from the Nibelungen Saga, killed the dragon Fafnir and became almost invulnerable by bathing in his blood. On your way up the hill you will encounter the Nibelungenhalle depicting scenes from Richard Wagner's operas.
The castle Drachenburg somewhat below the Drachenfels. Photo by Phantom3Pix under CC BY-SA 4.0
Schloss Augustusburg / Augustusburg Palace
Augustusburg Palace in the neighboring city of Brühl in the north of Bonn is listed as a cultural World Heritage Site by the UNESCO. The rococo palace and its surrounding gardens and parks are definitely worth a visit.
By direct regional train you will get from Bonn Central Station to Brühl in about 25 minutes, the Palace being in 5 minutes walking distance from the station.
While there, you may also consider visiting the noteworthy exhibition in the Max-Ernst-Museum (Dadaism and Surrealism), right next to the palace.
See the website for more information on Augustusburg Palace.
Photo by Wandernder Weltreisender under CC BY-SA 3.0
For more ideas, visit Bonn's tourist information website.