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)
Another possibility to achieve the A 3 in the direction Frankfurt is to take the train “RB 25” from the main station in the direction “Marienheide” or “Overath” for 3,20 €. Leave at “Stümpen” and turn right and go up to the “Kiefernweg” and turn right. Follow the street that is later called “Dammelsfurther Weg” up to the “Pestalozziweg” for maybe 500 meters and turn right. Now you pass the Autobahn and turn right again in the “Plantage”. Follow this way a few hundred meters through the forest and you will see a service station on the right side. Maybe you have to climb a fence. Now you are on the a station without oil station but much traffic. Easy to get a ride here. Totally it takes approximately 40 minutes from the cologne main station. In my experience this is the best possibility to leave cologne.
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 Strasse".
West towards Aachen, Netherlands and Belgium
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. You can reach this services by catching a train to Königsdorf Station and then walk down Augustinus Strasse and turn right to Marienhöfer Weg which leads you direct to the services. Alternatively, you can take Tram #1 to the terminus (Schulstrasse)in Weiden and walk ~4km along Aachener Strasse until Augustinus Strasse in Königsdorf.
North towards Düsseldorf or Dortmund
Take the tram no. 5 or 13 to Nußbaumer Str.. There is an onramp. A lot of traffic goes in your direction, but it is very dangerous for drivers to stop there.
When you arrive Cologne via the A1, it's difficult to get into the town center, because the motorway is very far outside. One option is to get off at the exit Köln-Niehl. There is a tram station, close to the motorway, in east direction, 15 minutes walking. At daytime, you can get to the town center in like 15 minutes.
From the same motorway (A1), you may as well get out at Köln-Lövenich and catch Tram #1 on Aachener Strasse direction downtown at either Bahnstrasse or Mohnweg stops.
Coming from the South on the A555 or through the A4, you may as well as the driver to leave you at the Am VerteilKreis roundabout situated just at the complete end of the A555, at the Kreus Köln-Süd (Junction A555-A4). The driver can really leave you at a petrol station and gain the motorway back with complete ease. From there, you can walk along the Bonner Strasse to catch a bus to the heart of Köln.