Luxembourg is the world's only sovereign grand duchy, and the capital behaves accordingly: calm, tidy and a little guarded. It also happens to be built on one of the most dramatic sites in Europe. In 963 Count Siegfried swapped some land in the Ardennes for a small Roman-era fort on a rock above the Alzette, and over the next 900 years the Habsburgs, Spanish, French, Austrians and Prussians rebuilt it so many times that it earned the nickname Gibraltar of the North. The old quarters and fortifications have been a UNESCO World Heritage Site since 1994.
The city is small, about 135,000 people, and it feels like a town that happens to have the European Court of Justice and a banking district. Three days is the right length: one day covers the old town, the casemates and the gorge, one covers Vianden Castle, the best day trip, and one covers the modern side on the Kirchberg plateau. Public transport is free for everyone across the whole country, so the day trip costs you nothing but time.
The plan is organized by place. Day one is the old town, the Chemin de la Corniche, the Bock and the Grund. Day two is Vianden, in the Ardennes. Day three is Kirchberg, the modern museums, the Pétrusse valley and, from late November, the Winterlights Christmas markets. The Mullerthal and the Moselle wine villages are in the more time section. Start on a Thursday: the Bock is open every day but the palace interior is not, Maison Victor Hugo in Vianden, MUDAM, the fort museum and Namur (all closed on Mondays) are open on the Saturday that closes the plan, when Winterlights is busiest. Coming from Germany, Cologne is a long but straightforward train ride away.