Weimar is nice and full of history. There is an airfield near by where you can sleep in small bungalows. Heidelberg is beautiful, too. You can go there by train from Mannheim Airport very easy. For Hamburg I can recommend landing in Uetersen, the public transport in Hamburg is great. Speyer has a nice town and the famous Technikmuseum in walking distance to the Airfield. Everyone of the Frisian Islands is worth a trip, although Juist has the shortest way to the Beach. Leipzig is great and Dresden, too. If you fly south along the Rhine, be sure to fly along the valley from Koblenz to Bingen where the Rhein features many old castles. Best flown low and slow. Münster has many churches and with Osnabrück nearby you have the most important cities for the end of the thirty years war. EDLT or EDWO would to the trick.
Fly down South, to Bavaria. There’s no GA airport in Munich but in Augsburg (40 miles) and Landshut (50 miles). I can help you get it organized.
Berlin, Hamburg, Dresden would be my other choices. Dresden is great for GA, and the city is beautiful. For Berlin you fly to EDAZ and take one of the rental cars which are offered in the Tower. There’s also a good hotel at the field, owned by the airport. I do that 2 times/year
Heidelberg hasn’t been mentioned yet… It’s within easy reach from Mannheim, which has an accessible airport but don’t waste your time in the city. I’m a fan of Heidelberg’s old town, though it is quite touristy and you will hear lots of English…
If Düsseldorf ends up on your list and you have spare time (probably not during a shorter trip like that) and feel like meeting up, I’ll be happy to. Düsseldorf is really good on a Wednesday night from very early spring to very late autumn on Ratinger Straße. It’s the nicest place to be among people. Virtually EVERYONE will be out there on the streets, sipping “Alt”, the local beer, chatting, near the Rhine..
Berlin, as others have mentioned, is a different league altogether and is THE place where stuff is just happening.
I just love euroga for the great answers one gets. Thanks to everyone and it seems sensible to wait a while until the weather warms up.
Try Speyer EDRY or Schwabisch Hall EDTY or Augsburg EDMA or Hamburg EDDH.
Sure, Lübeck is wonderful, but it’s something I would really try to schedule for the summertime instead. The airport is OK but not a highlight. I guess locals are a lot more critical about it.
Otherwise I agree with you, we all like different things about places and sure enough, Berlin is one of those famous examples.
Try Düßeldorf. Nice Old Town with great restaurants and good ambience (I’m actually there now and Karneval is on! 
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Fly in to Mönchengladbach which is very GA friendly and has ILS on both runways if you need it.
Where is your support for Lübeck, bosco? I thought that was your home town? I love it, very nice architecture and the Hanse history is very interesting and still very much alive. Of course Thomas’ Mann’s Buddenbrooks was my favorite book so I’m a bit biased.
It really depends what you think is pretty and interesting. Landscape, buildings, etc. or is it the flair? Berlin isn’t exactly pretty but it’s a vibrant place and it energizes people. It’s like New York in the 80s or Paris in the 70s, a place were things are happening. Strasbourg and Nancy are surely nicer than Berlin but they are rather boring compared to Berlin. Germany has the odd situation that its capital is actually the poorest place in the country with a lot of empty space. That brings a lot of creativity, a stark contrast to money places like London.
Hannover: go there when you have put a checkmark next to all other German cities. I think it’s mentioned in the book “1000 places you don’t have to see before you die”. Koblenz: go there as part of a Rhein tour but the city itself, nothing special really.
Not an easy one. Germany, particularly the west, is not renouned for its picturesque cities (someone sent a few thousand bombers over here some time ago which left most of the bigger and medium cities destroyed).
Koblenz is what we call a Beamtenstadt here in Germany; really nothing particular. The airfield is scenic but the city isn’t. Hannover has often been given the “most daft city of Germany award”. Yet the airport is fantastic for light GA.
Mannheim also has a good airport but the city is nothing really worth seeing. Totally destroyed in the war.
Frankfurt much the same, although it has a special flair due to its skyline, the Main river and its international character.
Just fly for 45 minutes and be in Strasbourg or Nancy instead. A striking difference.
Freiburg is really nice, and the airport is very close to the city. But maybe little too far for you.
The nicest cities are found way in the east IMHO. Erfurt and Dresden are wonderful, and they both have very good airports for GA. If you are looking for smaller places, look at Görlitz or Bautzen (both have good airfields as well).
But all in all, I would save Germany for summer. When it’s warmer, you can make cruise on the Rhein river for example. Or visit the islands off the North Sea coast.
But I have concur with Achim: go to Hamburg.
Go to Cologne this Monday and all you ever need in life to complete your carnival experience is Rio…
Hamburg EDDH is also reasonably priced for a large airport and it’s a fantastic city. Lübeck is one of my personal favorites in the north, also with a great airport for GA.
