If you are arriving in St Moritz on day one, I suggest doing the Lake Cavloc hike because it's not too long and can easily be done in a half-day. You have time to drop your bags at your hotel before heading out to the hike.
It's better to start in the late morning, so the shadows can fade and the cold corners can warm up a bit.
Try to be at Lake Cavloc at lunchtime and either have a picnic or eat at Restaurant Cavloccio by the lake, which serves local specialties.
Then continue the loop to the second lake. If you are traveling by public transport, make sure to check the bus schedule beforehand so you don't miss the last bus back to your hotel.
After the hike, I suggest stopping at the Silsersee viewpoint in my itinerary to snap some photos as the light starts to fade in the afternoon.
For dinner, I'd go to Hotel Waldhaus am See and get their "wild" menu. Best to reserve a few days ahead.
If you are driving to St Moritz from the north, we suggest driving over the Albulapass so you can stop at Lai da Palpuogna on your way. It is particularly beautiful in the early morning.