I have done this same route before, as part of my inter rail trip around Europe. This time I was taking my parents, and as it was later in May there was not as much snow, but still some snow on the high passes. We first took the train up from Lake Como to Tirano, then connected to the Bernina Express.

The Bernina Express train heads north crossing from Italy into Switzerland and climbing in elevation.

There was still fantastic views as it reached higher elevation where the snow was melting at this time of year. 

We took the Bernina Express all the way to Chur and stopped here for a couple of nights. Seen as we had rail passes we did a day trip to see around Zurich.

The following day we took the local trains that run along the glacier express route. This did involve changing trains a couple of times but is cheaper than having to pay for the reservation on the Glacier Express train. It was just as fun as we took our own lunch for the journey and enjoyed the views.

Near the end we switched trains and climbed up to the ski resort village of Zermatt, where we would stay for a few days.