The Lake Bike Rental shop is right Lake Moritz, a short walk from the St. Moritz train. Many nice trails leave from here. Or you can hop on the train with your bike to start your ride elsewhere.
They have variety of bikes for rent, including E-bikes and mountain bikes.
For kids, they have junior bikes with wheels from 16" to 24" and a small bike with training wheels. They also rent child seats.
Prices: You can rent for 1 -2 hours or for the day. Rates are surprisingly affordable for Switzerland - CHF 45/day for an E-bike.
Location: Via Grevas 34, 7500 St. Moritz (10 min walk from St Moritz train station)
Website: https://www.lake-bike.ch/
They have several suggested bike routes that leave from the shop. I would Lake Staz (1 hour), Val Roseg (2-4 hours) or Morteratsch Glacier (4 hours).
You can see the official Switzerland bike routes on map.schweizmobil.ch (blue are road cycling, brown mountain biking)
More Engadin biking info: www.stmoritz.com