As in most of Europe, indoor museums and galleries in Switzerland generally don't allow dogs, with occasional exceptions for very small pets in a closed carrier. Always check a specific museum's policy on its website before visiting if bringing a pet matters to your plans.
Outdoors, Switzerland is considerably more relaxed: historic old towns (Bern's UNESCO-listed old town, Lucerne's covered bridges, Geneva's old town), public gardens, and lakeside promenades are all generally leashed-dog-friendly with no special restrictions.
Mountain viewpoints reached by cable car or cogwheel railway — see the cable car guide for altitude exceptions — are a genuine highlight, and most are as welcoming to a leashed dog as they are to any other visitor.