From Sigeburg train station there is a S-Bahn (Cityrail) train into Bonn that takes about 20 minutes. So no taxi needed. That S-Bahn crosses most parts of the city, so you needn't necesarily get off at Bonn Hauptbahnhof (Central Station) if another stop is more convenient. Depends where your actual destination in Bonn is.

But I agree that the ride along the Rhine valley is much more beautiful. And cheaper, by the way. May take half an hour longer, though. Probably you'll have to change trains at Mainz.
www.bahn.de will give you all the information about train connections (click on "International Guests" if you prefer the English version).
To get connections along the Rhine valley, tell it to show connections from Frankfurt Airport to Bonn via Mainz.
Choose a seat on the right side of the train after Mainz if possible, as that is the side with the view (not possible when making reservations, but as soon as you get on the train you can look for a free seat).

Enjoy! I'm going along that route again next week, have done it several times, but I still enjoy watching the landscape and the castles.