The city's main museum and gallery sits directly opposite the cathedral ruins, which is appropriate given how much of its collection deals with Coventry's history of destruction and rebuilding. The History Gallery covers the city from the medieval period to the present, with good material on the Blitz and its aftermath. The 'Discover Godiva' gallery examines the legend of Lady Godiva — the real 11th-century noblewoman Godgifu, wife of the Earl of Mercia, around whom the naked-horseback-ride story accumulated over the centuries — and includes John Collier's famous Victorian painting of the ride.
The gallery also runs temporary exhibitions that are often better than you'd expect in a city of this size. Free entry.