Switzerland and the Bernina Express

Yes, you can stop off anywhere you want. And get another train later or on another day.
But the Bernina Express is just one train of many. All rail routes in Switzerland, including the line from Chur to St Moritz and the Bernina line have at least one train per hour, 7 days per week. The trains are a transport service for the locals, the Bernina Express is an extra limited stop train. Whichever train you travel on, you see the same mountains out of the window.

You can buy tickets (or use a pass if you are travelling enough to justify buying one) between any stations you want.

There are many lovely places in the Engadine (valley), all with stations. St Moritz is the largest, most well know and most expensive

There is a map showing all the Swiss rail lines here: https://www.swisspasses.com/railpass/overviewmap_en.pdf
And one just showing the Rhätische Bahn (the local railway company) here: https://upload.wikimedia.org/wikipedia/commons/7/76/KARTE_rhb.png

rockingrjranch, what would make it "worth it" to you? That is, what are your criteria for deciding its worth staying overnight?

What is your plan? Are you riding the Bernina Express to Tirano and then back the same day? Or to Lugano and then what? It is not clear at what point you plan this overnight in St. Moritz.

St. Moritz and Pontresina each have youth hostels as well as at least a couple of hotels in a "normal" price range. Booking ahead is probably your best bet to find something within your budget.

The actual Bernina Express trains have panorama cars and head sets in the seats with a multilingual monologue of the sights along the way. As such there is a separate "reservation" fee (about 10 to 20 CHF, varies by season) to cover these additional services. So if you ride the actual Bernina Express both days, you will pay the reservation fees twice. One advantage of the "standard" trains that run along these tracks, you can open the windows for better photos. Can't do that in the panorama cars.

Thank you all so much for your very informative comments and links. You have made my day!

We are planning on flying into Zurich and then the Bernina Express from Chur to Tirano, Tirano to Varenna and over to Bolzano to hike the Alpe De Siusi. We are then working our way down to Rome.

As far as St. Moritz I have no real reason to stop other than it looks beautiful and being in a luxurious playground. If there is a better place that is quaint or you think offers better sightseeing, we are flexible.

On our trip to Italy a year ago, we flew into Zurich specifically so we could take the Bernina Express route. We had been to Switzerland for four 2-week trips previously, but never made it as far east as the Engadine, and really wanted to see it. So I built in a two-night stop in Samedan, a small village very near St. Moritz but less expensive. We had a full day for hiking there and it was beautiful.

I do not know if Samedan is a stop on the "real" Bernina Express ( with special panoramic cars) but we rode regional trains from Chur to Samedan. After our hiking day we continued on regional trains to Tirano and then to Varenna. The advantage of the regional trains is they are more frequent ( so easier to schedule) and less expensive. We got a special price of 32 CHF for the two of us by booking well in advance.

rockingrjranch, no question its a beautiful area. St Moritz is an old-money luxury resort town, if that's your interest, but if you want to see the scenery and have nice walks, etc., any of those towns in the valley will do. Samedan and St Moritz are a short local train ride or a long level walk apart. There was a recent thread on the Engadin Valley you might want to Search for and read.

Thank you all so much for your helpful suggestions and comments.