Lucerne is the Switzerland that ends up on postcards, and for once the postcard isn't lying. A fast river leaves the lake under a roofed wooden bridge, a stone tower stands in the water beside it, the old town climbs a hill topped by a medieval wall, and Pilatus fills the skyline behind the roofs. The city has about 82,000 people and has been selling this view since the 19th century, when Queen Victoria came to see the Chapel Bridge and the Lion Monument and Mark Twain came to be rude about the souvenirs. He had a point, and the souvenirs are still here.
The honest character of Lucerne is a small old town and an enormous list of things to do within an hour of it. You can see the bridges, the wall and the lion in a day and a half. What fills the rest of five days is the lake and the mountains around it: a steamer across to the foot of Rigi, the steepest cogwheel railway in the world up Pilatus, a cable car through the clouds to Titlis. A bad-weather day costs you nothing if you keep the museums in reserve, and a good-weather day is worth a mountain.
The plan runs from the town outward. Day one is the old town and its two covered bridges, day two the lion, the Ice Age museum and the Swiss Museum of Transport, day three Pilatus, day four Rigi and day five Engelberg and Titlis. The three mountain days can be swapped in any order, so look at the forecast each evening and send the clearest morning up the highest peak. Start on a Tuesday if you can: the weekly market is on the river that morning and the family inn we recommend for lunch is closed Sundays and Mondays. Lucerne is also the easiest base for the rest of Switzerland: Zurich is under an hour by train, and Interlaken two hours on a scenic line.