Bernina Express - Which train can I take to get to Tirano from Salzburg to catch the Bernina express

Because of the lakes and mountains in that area, there is only one train line to Tirano. It comes from Milan to the south and west, and heads north into Switzerland. This means the Bernina Express is a good route between Italy and Switzerland, but not between Austria and Switzerland. You would have to go way south into Italy, or come from the north at Chur. Thus, the best way to do this would be as an out and back from the north, say from St. Moritz, Pontresina, or one of the smaller villages in the area (we liked Samedan).

I do not see how you can do all this in five days and have any time left for sightseeing in either Austria or Switzerland. I would drop Vienna, which is far to the east and out of your way.

I see a Geography failure here. The Bernina Pass rail line connects Tirano in Italy over the top of the Alps with the Engadine Valley in Switzerland (St Moritz etc.), and with the Albula line onward to Chur. The "Bernina Express" is a tourist train which runs Tirano-Chur, there are also hourly local trains following this route.
This is north-south across the Alps. From Chur you connect north to the rest of the Swiss rail network. From Tirano the connection is south to Milan and the rest of Italy.

Salzburg is north of the Alps, the quickest connection is to Chur. To get back north of the Alps from Tirano involves south to Milan, east across the Italian plain to Verona and then north over the Brenner pass, a very long way round. Look at a rail map, for example this one https://www.raileurope.com/europe-travel-guide/switzerland/map.html

"5 days to visit Austria and Switzerland."5 days should not be more than 2 places. Your plan is impractical.
I would do Frankfurt --> Salzburg (2 nights) --> Vienna (2N) --> Frankfurt and nothing else.

5 cities in 5 days. Why bother. You will only have time to check into a hotel at night and check out the next morning. You will see nothing of the cities you are visiting. I hope you are not thinking of getting a 5-day Eurail Pass and spend 5 days riding trains.

First, drop Vienna. Too far out of the way and no way to spend any time there to experience it.

So here is how I would do it (but I wouldn't because I would want to spend some time along the way).
Frankfurt to Salzburg.
Salzburg to Chur
Chur to Lugano using the "Glacier to Palms" Bernina Express train that includes a bus from Tirano to Lugano.
Lugano to Luzern, (don't bother with Zurich)
Luzern to Frankfurt

You might want to spend a day in Salzburg. in which case you can get trains from Lugano to Frankfurt in one shot.

Here is what the travel times look like if you omit Vienna:

Frankfurt to Salzburg, 5 hr 45 minutes. Spend 2 nights here
Salzburg to St. Moritz, 7 hours. Spend 2 nights here
St. Moritz to Tirano, 2 hours. Return to St. Moritz the same day, 2 more hours
St. Moritz to Frankfurt, 7 hours. (This is Day 5 of your trip)

I would find that a lot of train time. You could shorten it by going to Luzern instead of the Bernina Express. Spend two nights there and do a lake/cablecar/cogwheel train excursion on the intervening day.

Luzern back to Frankfurt is only 4 hours, so you would not need to depart until afternoon.

What time is your flight from Frankfurt on Day 6?

Thank you all for the replies. Truly appreciate you all taking time to reply. I think I understand now that I am trying to cram too many stops in 5 days. So I am thinking of just doing Salzburg and Vienna as per Chris F suggestion. I was just trying to get everyone happy because Salzburg and Vienna are my dream spots because I loved Sound of Music and I love music so want to visit Vienna. My husband and kids are not into it at all but are very excited to do the train hence thought we could do that for them. If I have to do just do Bernina express how can I plan that in 5 days? Thanks once again for helping me out with this dilemma. Also should I be bookign train tickets now if I have to travel first week of July?

Sam and Lola both described itineraries that would include the Bernina Express ( part or all) over five days from Frankfurt. Bot included Salzburg ( but not Vienna) so might be a good compromise?