Cologne, or Köln in German, is a big city in Nordrhein-Westfalen.
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.
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.
Not the best option unless you go to Bonn or Koblenz. For heading to Frankfurt, take the Bahn to last stop Koenigsforst (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 th A3, most of the cars will go to Frankfurt or can drop you at least on the Rasthof Siegburg (about 20 km south)