The Engadin / Graubunden canton is fantastic. Rick's suggestion for staying in Pontresina is right on. We found it had lots of charm and the people were friendly. We loved the Chesa da Mulin hotel; great breakfast, nice basic room, great location. The mid-day classical concerts (free) in the main church there are fantastic. High quality musicians and a great way to spend a few hours on a rainy day. We enjoyed the hiking around the lakes. Sils was fantastic, charming, picturesque, not to be missed.
For an out of the way, non-touristy experience, I recommend going to the Lower Engadine Valley. We saw very few tourists in our time there. Warning: one attraction, at the far end of the valley, is Tarasp Castle. It's interesting from the outside, very imposing on it's own tall hill. However unless you speak German, the tour itself will be a disappointment. There is a handout in English given at the start of the tour, but it is wholly inadequate. Also, be aware that the Engadine travel card given out by the hotels only covers the Upper Engadine, so if you want to visit Scuol, Zernez, etc., you have to pay on your own and it's not cheap. Still highly recommended.