Bernina Express rail line

No, those trains are wholly within Italy and are operated by Trenord (Tirano to Monza) and Trenitalia (Monza to Lugano). They are standard gauge, while the Bernina route is meter gauge. So you can buy tickets in Tirano.

The bus option is the official continuation of the Swiss Bernina Express route. The connection via Monza is using the Italian regional train along the east coast of Lake Como, almost all the way to Milan, and then doubling back along the main north-south rail line. You could consider staying on Lake Como (e.g., at Varenna) instead of Lugano.

I don't know why the regular trains timings are slightly faster. It doesn't seem to be the number of stops nor prolonged stops.

Thank you Sam and Laura for the clarification.
When I go to sbb.ch to look for this Bernina Express rail journey, it is no longer showing! Does it mean that it's been sold out or reserved?

Sorry, I can now get the schedules when I enter the two routes separately.
I assume that I don't have to make reservation for the first train route (from Pontresina to Tirano) because the trip is in May, correct? How do I make reservation for the bus route though?

On the train, you only have to reserve if you want the panorama car. The bus is reserved, since it's smaller and only running 1x/day. You should be able to get that at https://www.rhb.ch/en/panoramic-trains/bernina-express#cart.

If we were to take the train all the way from Pontresina to Lugano, I have a question from what Sam said: "So you can buy tickets in Tirano."
According to the sbb.ch website, the trip from Pontresina to Lugano is one whole journey with 2 stops at Tirano and Monza. Do I still need to buy a ticket for the leg from Tirano to Lugano or can I still use the Swiss Travel Pass for the whole journey?

According to the sbb.ch website, the trip from Pontresina to Lugano is one whole journey with 2 stops at Tirano and Monza.

I think you are misreading the SBB website. It is physically impossible to get from Pontresina to Lugano without changing trains.
I think you are confusing change of train with stops. The Bernina line ends at Tirano. Beyond that you have to change to the daily bus or another (Italian) train.
Option 1: Train Pontresina to Tirano (with intermediate stops), then once-per-day bus from Tirano to Lugano.
Option 2: Train Pontresina to Tirano (with intermediate stops), second train from Tirano to Milano (with many intermediate stops), third train from there to Tirano (with intermediate stops).

A Swiss Pass is valid on the daily bus from Tirano to Lugano (because it is run by the Swiss), but NOT on the Italian trains beyond Tirano. For option 2, you buy tickets for the train from Tirano in Tirano. Using the bus is a lot simpler.

Thank you Chris F for your detailed explanation!

As Chris notes, you would need to change trains (or to a coach) in Tirano. The Bernina Express track stops at the Tirano train station. The Bernina Express runs on a narrow ( 1000 mm) gage while I believe the Italians use a standard gauge track.

We road the Bernina Express from Davos to Tirano last week. The train stopped in Pontresina to switch engines.

We paid 45 CHF for a Rhaetian Railway day pass but had to pay an added 20 CHF for seat reservations. That covered us from Klosters via Davos to Tirano the an return to Klosters through the 19 km long Vereina Tunnel.

I believe that the narrow gauge and short radius of curvature combined with steep grades are the speed limiting factors of the panoramic trains. E.g Brusio spiral viaduct -- The spiral viaduct is 110 metres (360 ft) long, has a horizontal radius of curvature of 70 metres (230 ft), a longitudinal slope of 7%....

Edgar: Thank you for sharing your recent experience in riding the Bernina Express route.
Laura: We would like to take a standard regional train along the Pontresina-Tirano route rather than the official Bernina Express train with panoramic cars. I find that Rhaetian Railway does not allow me to just book for the bus but not the official Bernina Express train from the link you sent me.
Thank you everyone for your help/input in advance!

We would like to take a standard regional train along the Pontresina-Tirano route rather than the official Bernina Express train with panoramic cars.

Use the Swiss SBB route planner: http://fahrplan.sbb.ch/bin/query.exe/en?ld=std5.a&
The online timetable will only let you buy tickets a month in advance, but it gives you an idea as to what trains run when between Pontresina-Tirano. For example, for 24 March with arrival by 13:00, the route planner offers four options with arrivals from 11:00 to 13:00. Pick the one that is a "R" (Regio) and not the "BEX" (Bernina Express).

Thank you Edgar for your further assistance.
Laura: "On the train, you only have to reserve if you want the panorama car."
Yes, we don't want the panorama car, so we just need to reserve for the bus. But the link you sent me does not allow me to reserve for the bus only. Please advise!

You can reserve the bus at www.sbb.ch/en

Thank you Sam. I got it!!