By ship
By boat you can reach Königswinter with the tourist boats on the rhine, e.g. the KölnDüsseldorfer.And you can also reach Königswinter by ferry from Bonn Mehlem - but this is not quite useful if you're not living in Bonn.
By train
The nearest bigger railway stations to Königswinter are in Bonn, Cologne and Siegbur, but Königswinter also has its own railway station. It can by reached from Cologne by a regional train going to Koblenz. But make sure that your train is travelling on the right side of the Rhine, not via Bonn on the left, and that your train stops in Königswinter.
By S-Bahn
From these stations you have to travel by S-Bahn or tram. Long distance trains might pass Königswinter withot stopping. From Siegburg or Bonn you take the tram 66. Be careful that you do not use the train that is just going up to Ramersdorf.
You need a train going to Königswinter or Bad Honnef. From Cologne you can either take a tram or S-Bahn to Siegburg or Bonn or get a regional train that is directly going to Königswinter. Mostly, these are trains from Cologne to Koblenz but only those on the Rhine's right side. You you must look for trains passing Deutz, the airport of Cologne, Troisdorf and Beuel. Trains travelling to Koblenz via Bonn and Remagen won't bring you here.
By car
You can reach Königswinter directly from one of the biggest German motorway, the A3. Just take the exit "Siebengebirge" and follow the signs over the "Margarethenhöhe". If you are coming from Cologne you might also use the A 59 to Bonn and Neuwied. This motorway is also the right way for you if you are coming from Bonn. Just cross the Rhine and enter the motorway. Then you will have to take the exit "Königswinter". After entering the town signs lead you to parking places.