Lucerne Zermatt

Zermatt is at the end of a valley. Most activities involve going up the surrounding mountains (cable cars, mountain railways), looking at the views, and either hiking or sliding down again with planks of wood tied to your feet (aka "skiiing").

Luzern is a medium sized town / small city.
See this 360° panorama of Luzern: http://tour.luzern.com/360/en/

• Luzern has a nice old town you can walk around.
• The lake has boats you can go on. The boats are actually ferries travelling between the many lake-side villages. The lake is long, with many branches and at the far (southern) end fjord-like; very scenic. One option is to take a boat all the way to the far end (Flüelen) and get a train back from there.
• If you want to go up mountains, there are three near Luzern with cable cars / mountain railways to the top.:
  • Pilatus (the nearest): https://www.pilatus.ch/en/
  • Rigi (reachable by boat across the lake): https://www.rigi.ch/en
  • Titlis (train to Engelberg, then cable car): https://titlis.ch/en

When traveling between both places, make sure to go through the scenic Golden Pass Line. It is a very scenic railway.