The Royal Albert Memorial Museum and Art Gallery (RAMM) on Queen Street is one of England's best regional museums and entirely free. The building itself is a fine example of Victorian Gothic Revival, opened in 1868 and extensively renovated and expanded between 2007 and 2011.
The collections are wider than you might expect from a county town. The natural history galleries cover Devon's geology and wildlife with unusual depth. The world cultures collection is substantial — accumulated through Exeter's connections to global trade and empire — with strong holdings from the Pacific, Africa, and the Americas. The fine art galleries include a good collection of Dutch and Flemish paintings, and there are dedicated Devon collections covering archaeology, silver, and decorative arts.
Allow two to three hours to do it properly. The cafe is good. Open Tuesday to Sunday; closed Mondays.