Hi, and welcome to MySwissAlps!
>> We would leave Kassel in the evening of June 27 or early morning on the 28 to go to Italy via the Bernini Express.
From Kassel to Chur, from where the Bernina Express departs, is a journey of 7-8 hours, so leaving Kassel in the evening is not really feasible. The Bernina Express leaves Chur once per day at 8:32 in the morning and takes about 4.75 hours to get to Tirano. Regional trains following the same route from Chur to Tirano leave hourly throughout the day.
If you leave Kassel on the morning of 28 June, you might want to consider going through to St Moritz and staying the night there instead of at Chur. Although the trip from Chur to St Moritz wouldn't be on the official Bernina Express, you would be following the same route, and see the same scenery, as long as you put "via Chur" into the timetable for your trip from Kassel to St Moritz.
Then you could catch the official Bernina Express from St Moritz at 9:20 the following morning. Or even, better, spend two nights and a day in St Moritz if you have time, and continue your journey on 30 June.
>> We would like to know which is the fastest way/ direct/ and/or inexpensive way to get to Switzerland where we take the Bernina Expss which will take us to Italy ... I can imagine we would take a train in Germany that will take us to the Bernina Expss. What train/s? from where?
There is a direct train from Kassel-Wilhelmshöhe to Chur at 8:37. Other services involving a change of trains at Basel SBB and sometimes Zürich are also available if that time doesn't suit.
You can check your options in the Deutsche Bahn timetable here.
>> How to save on this part of the itinerary?
For the Kassel-Chur (or St Moritz) sector, Deutsche Bahn offers reduced-price advance purchase fares, but note that they are for a specific train, and cancellation/exchange fees apply. Full details are shown in the timetable.
>> While on the Ber. Expss., is it possible to have a stop or an overnight stop at a beautiful typical little country town and then hop again on the Ber. Expss.?
> services between May and October are limited to one departure per day from Chur, with one additional departure per day from Davos and St Moritz.
> boarding and alighting stations are restricted
> reservations are compulsory
So, if you break your journey, more than likely you would need to complete the journey on a regional train, which follows the same route, but doesn't have panoramic windows.
The timetable here has this information:
>> Is Tirano considered Italy
>> Should I make all train reservations from the States before we go?
Yes, if you are not using a rail pass of any sort, I would buy a ticket and seat reservation for the journey Kassel-Wilhelmshöhe to Chur (or St Moritz), and the Bernina Express from Chur (or St Moritz) to Tirano as soon as possible.
If you have a rail pass, e.g. Swiss Travel Pass or Eurail Pass, you will just need the compulsory seat reservation for the Bernina Express. Reservations are not compulsory for the trip from Kassel-Wilhelmshöhe to Chur, but I would recommend them for such a long trip.
You can buy tickets and/or reservations from the sites I linked to above for Deutsche Bahn and the Bernina Express.