Compton Verney is worth a look around. Loads of art & ancient artifacts (2000 year-old Chinese pottery etc..), and some interesting looking temporary exhibitions (there was a Van Gough one back in June). There's a £6 entry fee, but it's worth it for a good couple of hours of culture.
Further afield, Birmingham Museum & Art Gallery is brilliant; you could easily spend half a day there, and its FREE! (there are big boxes asking for donations - I always throw a couple of quid in as I always enjoy looking around). The restaurant there is really good too.
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