I tend to speed through museums and galleries but here is a “in process” list of those that have stood out in my travels. Not always the most famous, largest or well funded but I liked them.
Other than the Smithsonian Museums most are closed one day a week and smaller ones have even more restricted hours. Please check days and hours before planning your visit. Generally all museums, again other than the Smithsonian’s, charge an entrance fee. Many cities have bargain passes with reduced rates spread over one or more days to all or a large number of their museums and attractions. Check with the local tourist bureau or individual museums information number for further information.