We fly in to Zurich 4.5 days before our tour starts in Salzburg and I would like to make the most of it. The mountain scenery is my main interest, with gardens being secondary. I was thinking of going by train from Zurich to Berne, to sightsee in Berne for a few hours. Then we would jump on trains to take us from Berne to Zermatt, where we spend the night. The next morning (Day 2) see the sights available in Zermatt. I realize no time to do anything. Then jump on the Glacier Express for 8+ hours and end up sleeping in St. Moritz. The next day (Day 3) we jump on the Bernini Express to Tirano, Italy where we go by bus to Lugano, Italy where we spend the night. Day 4 we sightsee in Lugano for a very few minutes before getting on the Gotthard Panorama Express train and boat to get to Rigi Klum where we sleep. Day 5 we travel from Lucerne to Salzburg where we meet our tour group at 5 pm. I am a bit tired just from putting this together, so I think this plan may be too ambitious. But is it doable? Any recommendation for what should be cut out first? I am planning on using the Swiss Rail Pass (Flex).
That would be pretty amazing, but I think it is too much. It would be for me, and I love Switzerland. I suggest you shorten it by omitting Zermatt and the Glacier Express (my least favorite of the scenic rides) and make a lovely loop via the Gotthard route and the Bernina Express, incorporating Rigi Kulm and a charming little guesthouse in the Engadine 9nstead of staying in glitzy and expensive St. Moritz.
Take a look at the Swiss rail map with the scenic trains highlighted.
You can see the route, from Zuzern down to Lugano, over to Tirano (by bus), then the Bernina Express route toward Chur, with an overnight at Pontresina, or this charming little guesthouse which has its own stop (you may have to change to a regional train that actually stops there if you want to do this):
https://www.spinasbever.ch/en/home\
You route would be something like Day 1, Zurich to Luzern to Weggis by boat, cablecar up to Rigi Kaltbad and train from there to Rigi Kulm. Or spend the night in Rigi Kaltbad (that is what we did; there are several hotels, and the houses had lovely gardens when we were there in June).
Day 2, ride the cogwheel train down to Vitznau, and catch the boat down to Flüelen to board the train over the Gotthard Pass route to Lugano. Overnight Lugano. Note that the actuall Gotthard Panorama Express is a once-a-day run in First class only, but there are regular regional and IC trains on the same route. Unless you have a 1st Class Swiss Pass, or purchase an upgrade, you will want to take one of the other options.
Day 3, catch the Bernina Express bus to Tirano. and change there to the Bernina Express train to POntresina or beyond to Bever/Alp Spinas.
Day 4, ride the train to Chur, and from there make your way to Salzburg.