Independent extension recommendations for Germany trip

In which cities does your tour start and end? Give us an idea of the overall flow, and perhaps we can make some recommendations.

BTW, I think Leipzig is particuarly underrated.

We're starting in Berlin and ending in Prague (10 days total)

More time needed in Berlin. Extend in Prague too.
Bavaria is very scenic. Look at Obergammergau, Lake Contance,Garmisch/ Partenkirchen. Austria, Sslzburg easy drive from there.

The eastern part of Germany (former DDR) has a number of cities that survived WW II basically intact, retaining their glorious (and now restored) medieval architecture. My two favorites were Quedlinburg: with a castle, fabulous cathedral treasure (Google for American connection), small art museum, and English-language audio guides rentable from the tourist office; and Görlitz, which is on the Polish border: equally beautiful--movies have been shot there--even has some Art Nouveau buildings, and fewer tourists. I also really liked Erfurt, a larger university city with a beautiful historic district. The first two, at least, have over 1000 historic buildings.

All of the above are reachable by public transportation. I've never driven in Europe, but I believe aside from Berlin itself, there shouldn't be major problems driving in this area. I'd find trains (or buses if necessary) a more relaxing way to move around.

Rome2Rio.com will tell you whether train or bus service is available for any origin/destination pair you are considering.

But it sounds as if your choir will only be in Berlin for 2 or 3 days, with probably limited time to see the sights. Berlin is chock full of very good museums and historic sights (especially related to WW II and the Cold War). Before planning trips to other cities, I'd first read the "Berlin" chapter of a guidebook or two. I think you'll find that you want to spend some extra time there. Berlin (and eastern Germany in general) has comparatively inexpensive hotels. Stay as long as you can; it won't cost all that much.

After Prague: Franconia (region within northern Bavaria) is a fine destination dotted with some really lovely places. It's compact enough that you can stay in one base town and travel to the others, if you wish. These are all somewhat smaller places that unlike your other German destinations were not so severely damaged in WW II and that don't require a tour-bus overview - you can cover them on foot.

Bamberg, with its well-preserved medieval old town (a UNESCO World Heritage site) and brewery-restaurants.
Iphofen, a old-world wine town with an old town wall and towers, lots of artists and some Riemenschneider works.

Rothenburg, cutesy town that was half destroyed but nicely restored, gets overrun by millions of tourists every hear but is still worth seeing.

Bad Windsheim, wonderful open-air museum. Don't miss it.

Wuerzburg and Nuremberg are great places as well. The Residenz (palace) in Wuerzburg is especially worthwhile. W'burg and N'berg were both heavily hit and mostly rebuilt - but Nuremberg's old town zone and castle area are quite nice, and in addition to its other draws, Nuremberg's WW II historical offerings are quite good (if that interests you.)

Nuremberg is the transportation hub for Franconia. You can get there in less than 4 hours on the German Railways direct Intercity bus. It's faster and normally cheaper than the train options. Fly home out of either Frankfurt or Munich?

The highly-efficient German rail network serves all these towns. All except Wuerzburg are within the local transport authority (VGN) and can be reached on day passes from whatever town you stay in. A day pass for two called the Tagesticket Plus costs E18.70 and is valid all day long for two adults throughout the VGN rail network.