hey Neal,
Like Square already said: the weather's a mixed bag atm, temps are in the late 60s, cloudy and rainy. However the weather frogs claim a rise to 80 at the end of the week. So besides your fur Stetson you can also bring along the regular one
Here are some places to visit in case you want to skip Berlin:
Technikmuseum Speyer:
3 hours to the South, huge, probably worth a whole day trip, sub-related they sport a type 205, and some small boats like Biber, Neger and Seehund, scenic route (for an Autobahn)
Rheinmuseum Emmerich:
The closest submarine from here, a Biber, tiny museum about Rhine shipping but only 1h away
Wehrtechnische Studiensammlung, Koblenz
(link to the wiki page because the museum's website is too horrible

- mosly focused on Bundeswehr weapon systems, they have a Seehund, 1.5h, beautiful city
Liberty Park (National War and Resistance Museum), Overloon, Netehrlands:
no subs here, but tons of stuff about the Allied efforts in WW2, also only 1h
Marinemuseum, Den Helder, Netherlands
The Tonijn (Potfis-class) lies here, 3.5 hours, located at the coast, many ships there, they call it Museumharbor
Deutsches Marinemuseum, Wilhelmshaven
3.5h to the North, about the history of the German Navy since the 19th century, A Seehund is there. Special exhibit about Schellboote (PT-boats) atm
Not far away, on the other side of the bay:
Deutsches Schiffahrtsmuseum, Bremerhaven
This is the one where the Wilhelm Bauer, the last Type XXI, is located, also 3.5h
Hürtgenwald Museum
about the Hürtgenwald Battle in 1944, small museum, 1.5 h, scenic route to ride, only open on Sundays (?)
If you are into industry culture, there are many places to visit in the Ruhrgebiet, which was built on coal and steel, half an hour North of here
http://www.route-industriekultur.de/
For WW1 history, there are tons of places, mainly in Flanders (belgium), with Verdun about 4h away.
If you just want to take a bike tour to the countryside, I'd recommend either go to the Eifel region, or very pittoresque, take a ride along the Mosel river. The road goes just along the river, several hundred feet above it, with great views just at very corner
2 more driving tips: Be aware that we have an insane amount of trucks on the roads nowadays, many in shabby conditions (vehicles and drivers). Another nuisance any American driver notices are the billions of traffic signs - they grow here like kudzu in the Southern States