Historylinks is a small, well-run local museum near the cathedral, with a five-star rating from VisitScotland. It covers the town from prehistory to the 20th century: the cathedral and the bishops, nearby Skelbo Castle, war memorials, the Janet Horne witch trial, and of course golf.
It's open from late March to the start of November, 10:30–4, and visits outside the season can be arranged with a day's notice. It's an hour well spent before or after the cathedral.