Lucerne to see Switzerland

Three nights means only two full days. Luzern warrants at last one day to explore and walk around the town from Lion monuments to old town and the bridges. Down load the Luzern city guide for walking tour map:

http://www.luzern.com/en/brochures

Boat ride and mountain day excursions:

https://www.rigi.ch/en/Experience-Enjoy/Summer/Summer-experiences/Classic-Rigi-Round-Trip

http://www.pilatus.ch/en/railway-cableways/panorama-gondola/
http://www.myswissalps.com/pilatus/gettingthere

The route

Section 1: by train or boat from the rail station in Lucerne to
Alpnachstad;

Section 2: by cogwheel train to the peak of Mount Pilatus;

Section 3: by cable car down to Fräkmüntegg;

Section 4: by gondola down to Kriens;

Section 5: by bus back to the rail station of Lucerne.

1) Boat trip on the lake, either a circular trip, or one way and a train back: http://www.lakelucerne.ch/en/
2) On whichever day has better weather, go up Titlis (train to Engelberg, then cable cars): https://www.titlis.ch/en

Luzern is a lovely city, and worth ½ day exploring.