Help with planning excursions around Lucerne and Zurich

The Tell Pass is a rather expensive option at 230 CHF pp for 3 full days, but is covers a lot, all rail, boat, bus, mountain lifts in the region.
https://www.tellpass.ch/en

One common excursion is boat to Vitznau, then Rigibahn to the Rigi mountain top, then hike downhill a short way to Kaltbad and take the cable car down to Weggis, then a short downhill walk through town to the dock for the boat back to Luzern.

Another is a trip to Mt Pilatus, combined with a boat ride and rail line to the peak and a cable car ride down the other side and bus back to Luzern.

You can also do a day rip to Engelberg to see the high Alps with several mountain lifts, including Mt Titlis. The Tell Pass also covers the scenic train ride to Interlaken and back.

Zurich has all-zone day tickets (24 hour) for 35.60 CHF.
https://www.zvv.ch/zvv/en/travelcards-and-tickets/tickets/daypass.html

We spent two nights in Engelberg, then three nights in Lucerne. Engelberg was really a delight - a small town feel, and the opportunity to take 3 cable cars up, up, up to Mt. Titlis. We did this in July, and it was so wonderful and spectacular to be in the snow at the top of the mountain in summer! It was also interesting to see how people reacted to their first snow experience in their lives!

I would highly recommend this trip, and on the way back down, get off the cable car at Trubsee and walk down to the lake. It is beautiful, and when we were there, we found ourselves in the middle of a cow parade! There were rowboats and hammocks available on the lake. It felt like a wonderful surprise to us to realize we could get off the cable car and have this pretty side trip before returning to Engelberg. It was a fairly short train ride from Engelberg to Lucerne.

Laurie

Of course, you can easily spend a relaxing day in Luzern itself, walking the (partial) city walls, climbing the 3 city towers, marvelling at the two wooden bridges, visiting the Art museum (Rosengart) or the humongous Museum of Transportation, viewing the stone Lion of Luzern, visiting the Rathaus Brewery for an outdoor lunch, etc.

Hello, thank you all for your input. Greatly appreciated!
I tried with ChatGPT to come up a plan :), and below is the summary of what it gives. Does it sound reasonable or is it too much? And based on the itinerary, it also recommends Swiss Travel Pass. Is it a good idea?
Thank you very much. :)

Day 1: Arrival in Lucerne-- Explore Lucerne:
Day 2: Lucerne & Mt. Pilatus
• Morning: Take the world's steepest cogwheel railway from Alpnachstad to Mt. Pilatus (Pilatus Kulm). Enjoy the panoramic views.
• Afternoon: Return via the aerial cableway to Kriens and take a bus back to Lucerne.
Day 3: Lake Lucerne Boat Cruise & Rigi
• Morning: Take a boat cruise on Lake Lucerne to Vitznau.
• Midday: Ride the cogwheel train from Vitznau to Rigi Kulm (Mt. Rigi)
• Afternoon: Descend to Arth-Goldau by train and return to Lucerne by regular train.
Day 4: Day Trip to Engelberg & Mt. Titlis
• Morning: Take a train to Engelberg.
• Midday: Ride the Titlis Rotair, the world's first revolving cable car, to Mt. Titlis. Experience the glacier, ice cave, and suspension bridge.
• Afternoon: Return to Engelberg and explore the village before heading back to Lucerne.
Day 5: Travel to Zurich & City Exploration
Day 6: Day Trip to Lauterbrunnen
Day 7: Old Town (Altstadt), Lake Zurich, National museum and Uetilberg late afternoon

Skip the day trip to Lauterbrunnen from Zurich. From Luzern it is a lot closer and a more scenic train trip.