Reading Museum, housed in Reading Town Hall, is free and covers the town's history from the Roman settlement of Calleva Atrebatum (Silchester) through the medieval abbey to the Industrial Revolution biscuit empire of Huntley & Palmers. The Roman gallery, with finds from the nearby excavations at Silchester, is the strongest permanent collection — mosaics, sculptures, and everyday objects from what was a substantial Roman town.
The Bayeux Tapestry replica, displayed in a dedicated gallery, is the other major draw: a full-scale Victorian embroidered copy of the original, made in the 1880s and displayed in proper sequence so you can follow the entire narrative from Harold's oath to the Battle of Hastings. It is a remarkable object in its own right. The museum also holds changing exhibitions and is worth checking for programme details. Entry is free.