The Bergmancenter, honouring Sweden's most famous film director, the late Ingmar Bergman, a Fårö local, was inaugurated in 2014 and hosts exhibits, lectures, workshops and screenings, as well as a library and cafe. It's free to take a look around the centre but prices vary for screenings and special events.
The centre can set you up with your very own Bergman guide of the area (125kr for one hour) or you can try a Bergman bus safari (tours every other Saturday in July and August from 3pm to 5pm, from 325kr) or during Bergman Week in late June.