Okay, here's a few ideas (maybe more than a few, lol)!
Aachen: I loved Aachen. Its cathedral is quite beautiful (especially the interior, which is drop-dead gorgeous) and houses the remains of Charlemagne, leader of the Holy Roman Empire (they are in a gold shrine that hangs towards the back of the altar). It also has the Barbarossa Chandelier, which was made on the order of Emperor Frederick I sometime between 1165 and 1170 and was installed under the cupola of the Palatine Chapel in Aachen Cathedral. It's really beautiful.
The town is also very charming and has a lovely vibe to it. It's smaller than Cologne and you could easily spend some time just walking around and enjoying the sites. Don't miss the Granus tower, which is next to the town hall and is one of the few pieces remaining from Charlemagne’s palace. It dates back to the 8th century, making it over 1200 years old. I had a really nice Apfelstreuselkuchen and cappuccino at Café zum Mohren.
Münster: First, don't miss Saint Lambert Church in Münster. This gorgeous church was built in 1450 and has a number of beautiful parts to it. Something not so beautiful are the three cages at the top. In the early 16th century, several Anabaptists were captured, then tortured and executed. Their bodies were placed in three iron baskets that sit high among the steeples of the church. The bodies were eventually removed 50 years later, but the cages still remain.
Also, there is the Rathaus (town hall). In the 1600s, it was the site of the signing of a peace truth ending the 30 years war. A similar treaty was signed in nearby Osnabrück. Osnabrock was very Protestant and Münster was very Catholic, so both sides were happy and felt that they could maintain some control over the process.
Munster Cathedral is very lovely, and don't miss the astronomical clock. It's so cute—at the very top, there is a little circle of figures that come out of a little room and then go into another. The door shuts, and then they come back out, and so on. It was a lot of fun to watch.
Cologne: Of course you can't miss the cathedral, which is drop-dead gorgeous! But Cologne is really a very nice and walkable city. I liked it a lot. You could also check out the Hohenzollern bridge, which is decorated with thousands of love locks.
I had a very memorable meal in Cologne at Maibeck für Dich. It was one of the best meals I had in Germany. Not German food, but just really good food. I also liked Brauhaus früh am Dom, which serves German food. I had the sauerbraten, which was very good. Both restaurants are within walking distance of the cathedral.
Regensburg: Regensburg was really a charming place, and of course, the town has a gorgeous cathedral. Another interesting spot is Porta Praetoria. You can find it by exiting the cathedral at the north transcript from the cathedral. This is Germany’s oldest stone structure and is the remnant of a former gateway dating from 179 AD that once led into a Roman military camp called Castro Regina, and is almost 2000 years old.
The tourist information office used to give guided tours of the Town Hall for a small fee, and they had some in English, so you might want to check it out. I really enjoyed it, as it included some very historical rooms and even access to the dungeon, where there was an authentic torture chamber. One restaurant I liked in Regensburg was Café Orphée, where I had a lovely brunch of meats, cheeses, bread, fruit and butter and marmalade, along with a cappuccino. There is also the house of Oskar Schindler (Watmarkt 5), which is a private building where Oskar Schindler lived briefly after World War II. It's not open to the public, but there is a plaque in front that was interesting to see.
