My favorite way to see Mount Rigi is by taking the ship on the lake, from Luzern to Vitsnau, where I catch the cog-wheel train from the ship dock (across the street) that goes up the mountain with a stunning beautiful view of the lake (out the left window). To get back to Luzern, I like to take the cable car ride for amazing views from the air, down to Weggis. At Weggis there's a long downhill sidewalk to get to the lake and ship dock, for the boat ride back to Luzern.
(I didn't find the train route from Luzern to Arth-Goldau to Rigi Kulm to be very interesting in comparison. The railway from Arth-Goldau seemed to me to be more of the back way there.)
On Mt. Rigi, you can go to the top (Rigi Kulm) and then walk a path down or take the train down to Rigi Kaltbad. You can take the Blumenpfad (labeled flower path) from Kaltbad either way from the cog-rail station for exhilarating views of Rigi and Lake Lucerne below. (Actually, if you walk down to Rigi Kaltbad from Rigi Kulm or Rigi Staffel, you can join up with the Blumenpfad at the viewing platform before you reach Rigi Kaldbad.)
The cable car ride to Weggis starts at Rigi Kaltbad.
The Swiss Travel Pass fully covers the lake ship, the cog-wheel train, the cable-car, and the train from (Luzern and) Arth-Goldau. (In other words, all of Rigi public transportation.)