When most Americans think about Germany, they picture Lederhosen, buxom barmaids, and picturesque mountains. That's found mostly in Bavaria.
There's a joke that Bavaria is kind of like Texas: big states, largely rural, more conservative and religious than other states (though Bavaria is mostly Catholic, while Texas is protestant), would secede into their own nation if possible. Bavaria is formally known as the Free State of Bavaria, and in the wars of German unification, led by Kaiser Wilhelm I of Prussia, the Bavarian kings resisted and sided frequently with neighboring Austria over their more aggressive northern neighbors, unless long-standing rival France was involved.
The Thirty Years' War raged across Bavaria in the mid-1600s, pitting the Protestant north against the Catholic south.
Nuremberg is a lovely town in north-central Bavaria. The imperial castle overlooks the city, and you can walk around the former moat, now dry. Probably most famous for the trials of war criminals held there, Nuremberg also has the Plaza of the Declaration of Human Rights, which has each of the articles engraved on a pillar in multiple languages. Nuremberg has a lot of very strange statuary throughout the city, including a creepy dead rabbit and an allegory of Death.
One of the coolest things to do in Nuremberg are the Felsengänge, a series of interconnected tunnels under the city, which were built in the 14th and 15th centuries as beer fermentation cellars and were used as air raid shelters during World War Two. As much as 75% of the city center was destroyed in Allied bombings, but Nuremberg had a lower death rate than similar-sized towns because of the beer tunnels. ("Das Bier hat uns das Leben gerettet.") Check the website for times and prices and advance ticket purchase. Highly recommended.
On the topic of Allied bombings, Sankt Sebaldus Kirche was largely destroyed, except for the main supports on the corners. When I was there in 2005, they had a display of immediate post-war pictures inside. Some of the original sculptures and windows remain, damaged, of course. I found it very moving, but I'm predisposed to the rebuilding from rubble/ashes type of story.
The Romantic Road runs through a lot of quaint towns in Bavaria. For the fit and energetic, it's possible to bike the length of it. The rest of us can take the train. One of my favorite stops on the Romantic Road is Rothenburg ob der Tauber, the only city, according to locals, with its medieval wall largely intact. (Rebuilt, naturally, after having been bombed to smithereens.) Rothenburg has an enviable position on a hilltop with a cliff on one side and the Tauber river on the other. It was also at the junction of several trade roads, but after the Thirty Years' War was broke. The Black Death didn't help matters any, either. When other cities were taking down their walls, Rothenburg didn't have the money to follow the fashion. This would work to their advantage several hundred years later, when romantic poets and painters went to visit and told their friends about this quaint walled city in southern Germany. There's a lot of kitsch, but it's worth a visit. You can see everything there is to see in a single day. Take a trip up the city hall tower to get a view of the city, and walk along the wall. You can take it almost the entire way around. Get a Snowball at Diller's. Take a tour with the Night Watchman. Have dinner and a drink in Hell. Get your Christmas ornaments at Käthe Wohlfahrt.
The Romantic Road ends in Füssen, your starting point for a trip to Neuschwanstein Castle. The town itself is scenic and has some typical German architecture, as well as restaurants to feed you after a nice hike back down from the castle itself. It's an easy day-trip from Munich, and fully accessible by public transportation. Buy your ticket at the ticket center before hiking up the mountain. It will give you an entry time/group number, and you wait in the courtyard of the castle for your group to be called. Neuschwanstein is the most touristed place in Germany, for good reason. When Mad King Ludwig was building the castle in the 1880s, Pöllat Gorge was known as a romantic hiking and picnic spot. A bridge behind the castle crosses the gorge and gives you a great photo op of the castle.
If you want to hike the gorge (good weather only!), do it on the way back. You might miss your tour if you hike up.
Do I need to say much about Munich? No tour of southern Germany is complete without a stop in the city most Americans have heard of. From there, you can go to Neuschwanstein, as I mentioned above, and Dachau. In central Munich, the Altstadt, you can visit the Viktualienmarkt (a huge open-air market that sells victuals, largely sausage, bread, cheese, and fruit, as well as souvenir folk art), Marienplatz (where the Gothic town hall with the famous bell tower lives), towers, cathedrals, and various statuary. Outside the city is Nymphenburg Castle, where you can find some particularly mean swans in the water gardens. The German Museum (Deutsches Museum) is home to some seriously cool technological artifacts: original steamboat engines, a U-boat, an ENIGMA machine, and several really old computers.
The train ride from Munich to Füssen is gorgeous. Off to the south, you can see the Bavarian Alps. It's rural, so you have to take a regional train, which stops at every little burg on the way.
There's a lot more to Bavaria than this handful of places, but they're the big ones, and I've spent a lot of words on this handful. Next up: a trip back west to the Rhine & North Rhine-Westphalia.