Transport to the start of the race was very good last year - extra trains coming in from all over Graubunden very early in the morning for it. There should be details of the schedules on their website. We weren't sure about the tiny station that we needed to use, so we called the RhB (local train company) a couple of weeks beforehand, they got the info for us, and it all worked out fine - train to St Moritz, then shuttle bus to the start.
They have a fleet of army trucks to transport your gear from the start to the finish area, which also looked very efficient (although we didn't actually need to use it):
There is a train station right next to the finish area which is only used for the race, not during the rest of the year. In the finish area, they hand out colored tickets that correspond to which train you'll be able to board, and then make announcements when people with a certain ticket color should head towards the station, where everything was again very well managed.