I just reviewed the thread.
Here are a couple more thoughts.
1.- You express an interest in seeing the Swiss Alps.
The Rigi at about 1800 meters has good hiking and good scenery on a clear day. However, it does not provide a high Alpine experience. It is a kinder and gentler mountain, with the peak (Kulm) just barely above the tree line.
Altitudes on this map:
map.geo.admin.ch/?topic=ech&lang=en&bgLa yer=ch.swisstopo.pixel karte-farbe&layers=ch.bav.ha ltestellen-oev,ch.swisstopo.swiss tlm3d-wanderwege&layers_visi bility=false,false&E=2 672880.00&N=1208110.00 &zoom=5
2.- Two other options near Luzern are Titlis and Pilatus.
Pilatus is quite high and rocky and accessible easily from Luzern....going up via Kriens early in the day to avoid August crowds on the cograil; returning to Alpnachstad to ride the cograil down and a lake boat before the crowds come back down.
Lake boat connectiona across the lake from Vitznau to Alpnach ( or vice versa) exist but are infrequent:
There is not much space or hiking up top.
Titlis provides a true Alpine experience, with some snow probable even in August.but is a bit more difficult to reach from Vitznau than from Luzern. Use the timetable.
3.- The best Alpine experience within reach is the Jungfrau region.
If you don't actually go up to Jungfraujoch, but confine your journey to Schilthorn or to the rail loop through Kleine Scheidegg, itis possible and not so expensive as the Jungfraujoch train.. There are lots of choices and options for trails, cableways, villages, etc. It is a tolerable but long day trip trip with return from Luzern on a scenic rail line. Going from Vitznau adds significant time to the journey.
If this is truly a once in a lifetime experience to see the Swiss Alps, I'd give it careful thought.
Rooms in Luzern in August are hard to get.