Cologne, or Köln in German, is a big city in Nordrhein-Westfalen.
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South towards Bonn, Koblenz, Frankfurt am Main
To get out of Cologne to the South the best possibility is to go to the Verteilerkreis Süd, a big roundabout where all vehicles that are heading for the motorway to the South or West are passing by. There is also a petrol station beside it and a traffic light where one can ask people for a lift.
Not the best option unless you go to Bonn or Koblenz. For heading to Frankfurt am Main, take the train to the last stop Königsforst (nr 9, may have changed though), walk straight for about 15 min. Then after a right turn you are on a decent on ramp of the A3, most of the cars will go to Frankfurt am Main or can drop you at least on the Rasthof Siegburg (about 20 km south)
East towards Berlin
the first service area on is rasthof remscheid. you can get there by public transport (quite inconvenient though) or try one of the on ramps in the north east of cologne. i think the best one is close to KVB-station "Rixdorfer Str."
West towards Netherlands
On the motorway ring around Cologne, to the west of the city, there's the motorway service area Köln-Frechen, Germany's busiest service station. There you can get lifts to the Netherlands and Great Britain from one of around 200 Trucks that are there in average.