Dresden or Hamburg

Bill,

I haven't been to Hamburg, but I would certainly recommend Dresden. I spent a couple of days there, and was wishing I had booked more time, as there's lots to see.

It depends on your interests. Hamburg is a modern, posh port city, Dresden is very "Old Europe" with a small reconstructed old town. In Dresden you would probably mostly visit museums, in Hamburg the huge port and Miniatur Wunderland.
I would recommend neither, since you will just see more of the same. Hamburg is a modern post-war city like Berlin, and you will already see more than enough museums in Berlin. The trips I would recommend, which all feel very different from Berlin, are:

Potsdam
Potsdam is much more than just Sanssouci. It was the summer residence of the Prussian kings for more than 200 years, and to do the city justice you need at least 2 full days for Sanssouci, the old town and all the other parks like Babelsberg, the New Garden or Peacock Island.

Spreewald
If you like nature you will love this. The Spreewald is a region where the river Spree meanders in hundreds of small water ways through meadows and forests. And it's the home of the Sorbs, a small Slavic minority. You can do tours on punted barges, rent a canoe, hike or bike. The perfect place to relax.

Quedlinburg
This is probably the best preserved medieval town of Germany, with thousands of half-timbered houses. Three hours away from Berlin (direct train on Saturdays AFAIK). From there you can do day trips to other equally beautiful towns like Wernigerode or Goslar or take a nostalgic steam train up to the highest peak of Northern Germany, the Brocken.

Schwerin
This was the seat of the duchy of Mecklenburg since the Middle Ages. Beautiful preserved old town surrounded by lakes, and a fantastic and picturesque fairytale castle on an island.

Erfurt
The capital of Thuringia offers no real highlights, but one of Germanys biggest preserved old towns. From there you can easily visit Weimar and Wartburg castle, both World Heritage Sites.

Other worthwhile destinations are:
Lübeck which impressive history as a major port city still shows (but 3 hrs from Berlin)
Leipzig which I found amazingly attractive (and only some 1 hr from Berlin)

Thanks, great recommendations I didn't consider.

",,,in either Dresden or Hamburg." Ideally, you should do a day trip in both cities. Given this particular choice, I suggest Dresden. Hamburg is great too but demands more than a day trip. An overnighter and day trip in Dresden can be done if the focus is only on the famous sights.