travel from Lugano to St Moritz

There are two ways to do this. There is a Swiss Post bus that runs once per day from Lugano to Tirano, connecting with a train over the Bernina Pass line to St Moritz, or another daily bus direct from Lugano to St Moritz.

Lugano, Stazione Nord depart 12:05 BUS 631, St. Moritz, Bahnhof arrive 15:45

Lugano, Stazione depart 10:00 BUS 5950, Tirano, Stazione arrive 13:00
Tirano depart 13:40 train R 1656, St. Moritz arrive 16:11

These are special buses, and I think they need reservations.
Other than that, you can go Lugano --> Belinzona --> (bus) Thusis --> St Moritz.
All three routes are scenic. You can look them all up on www.sbb.ch

Very helpful. Thank you

Hi - Husband and I are traveling in October from St. Moritz or Pontresina on Bernina Express (or regional train??) to Lugano. Any comments or recommendations? I'm getting a little confused. Have first class Eurail passes. thanks!

If you are doing the Glacier-to-Palms, that is the Bernina Express train to Tirano then the Bernina Express bus to Lugano. A 14 CHF pp reservation fee is required in addition to the Eurail pass, valid for Switzerland and Italy, I presume.