Switzerland is part of the Schengen area, so there's no routine passport control driving in from an EU neighbour, but Switzerland is not in the EU customs union, and occasional spot checks do happen at road crossings. Keep your pet's passport, Swiss pet passport, or health certificate accessible in the car at all times, not packed away in a suitcase.
Inside the car, the same general safety practices apply as anywhere in Europe: restrain the pet with a harness, carrier or dog guard, stop every couple of hours on longer drives, and never leave a pet alone in a parked car, especially in summer.
If your route crosses an alpine pass, check seasonal closures before you go — some high passes close completely in winter and reopen only in late spring, and route via a tunnel (like the Gotthard) instead. Winter tyres or chains are commonly required by law from specific dates depending on the route and canton.