Somerset:
Bristol's City Museum and Art Gallery - outstanding & diverse range of objects
Somerset Rural life museum - Abbey farm, Glastonbury
Ham Hill Country Park  - Public open access all year round.
Langport & River Parrett Visitor Centre  - River Parrett trail & ‘discovery room’.
Blue Anchor Railway Museum - small museum open Easter until September.
Perry's Cider Mills - working cider farm & rural museum.

Devon:
The Bill Douglas Centre - for the history of cinema & popular culture. Exeter
Royal Albert Memorial Museum & Art Gallery - Exeter city centre.
Tarka Trail & Bideford Railway Museum - Traffic free and on the level, this section of the 180 mile "Tarka Trail"  runs betwixt the B3233 and the River Torridge on what was once the Old Bideford  to Barnstaple Railway line.

Dorset:
Corfe Castle Museum - housed on the ground floor of the smallest Town Hall in England, so it is quite small too.
RNLI Headquarters and Museum, Poole - heritage display of the history of the RNLI; 1824 untill present day.
Tolpuddle Martyrs Museum - tells the harrowing tale of the Martyrs which lead to the foundation of modern day trade unions.
Water Supply Museum, Weymouth - also a spring source and current day pumping station.