Welcome to Switzerland!
As a general rule, you must buy half fare tickets for your dog, similar to what a child (6-16) or people who own the Half Fare Card need. However, there is also a dog day pass at CHF 25/day. When all your journeys in one travel day exceed the total cost of CHF 25, you buy the Dog Day Pass - it gives free admission on all public transport (cableways may be different). If you do a short ride only (say, you’d spend less than CHF 25 on half fare tickets per day), you can buy a point-to-point half fare ticket.
Kindly note: dogs are not allowed on passenger seats and have no admission to dining cars.