UthpaleK Can I definitely get in to any train on the day without a pre booking?
Railways here work on a “proof of purchase” system. So in order to travel on a train you need to be able to prove, when requested, that you have paid for the right to travel on that train.
There are no pre bookings, no check ins, no ticket inspections at the station etc… You just get on a train (or bus or tram) and when requested by a ticket inspector or conductor just show your pass. Buying a saver day pass you have bought the right to be on board any public transit vehicle in Switzerland for a whole day.
Note, if I were you I would not use the SDP to go to Interlaken and Grindelwald. Once in Grindelwald you will have to pay full price for any cable cars.
What I would do is visit the Rigi. That is a mountain railway that is covered by the SDP, and you can combine that with a boat trip on Lake Luzern for example.
Or go to Mürren. Both the cable cars to Mürren are covered. Plan your day around getting the most out of the SDP.