While you could take the direct train to Cologne and then the train from Cologne to Bonn, that's basically a waste of time. I'd recommend you take the shuttle bus from the airport into Bonn directly (line 610); it'll lead you right into the center of town. In case you wonder why there is no decent train connection from the airport to Bonn: The airport is on the "other" side of the Rhine, and the next train bridges to cross the river are in Cologne and near Koblenz.

In terms of things to see, a lot of the "essentials" have already come up. If you are into having nice walks, I'd recommend to not just walk up the Drachenfels (that's the tourist's standard) but explore that general area a bit ("Siebengebirge").

Other things to do include having a look at the remains of the bridge at Remagen; visiting Adenauer's house nearby (which is now a museum); walking around Bonn's old town and looking at the Palace that the archbishops of Cologne built there (now the University's main building), and the smaller one in Poppelsdorf (University-owned, too).

Finally, to get from A to B in the city center, walk or use the (excellent) public transportation. Don't drive -- the street system is a crazy maze of one-way roads that leads even locals into desperation.

PS: Bonn is a pretty good basis to explore Cologne, too. The city centers are less than 30 min apart if you take the train.