Barcelona is capital of Catalonia and world-famous for soccer and architecture. You must visit the cathedral Sagrada Familia, designed by Gaudi, that has been under construction since 1883. Camp Nou, where FC Barcelona resides, is an experience of a different kind.
Madrid, capital of Spain and home of the royal family. The city center with great architecture is amazingly small, with nice places to enjoy local food and wines and very walkable.
Seville is the capital of the Andalusian region that also includes Malaga and Granada. It is very Spanish here with historical buildings, Flamenco Music & Dance, horses, Paella and Sangria. Note that it can be really hot (40°C and more) in Summer...
Granada is a major tourist destination in Andalusia because of the Alhambra. The nearby Sierra Nevada mountains (3000+m altitude) allow for skiing in winter, with a unique view of the Mediterranean.
Cordoba has a beautiful old city center with a lot of charm. The main attraction however is La Mesquita, a cathedral that started as a mosque. Cordoba is one of the hottest cities in Europa in Summer.
Valencia is an import coastal city with unique architecture, including a beautiful market hall. It is also home to the Paella de Valencia, the original!
Segovia is located just north of Madrid and very well worth a day-trip. The main sights are the old city center with large Roman Aquaduct, the Cathedral and the Phantasy Castle by Alfonso II.
San Sebastian is a superb beach location just west of the Pyrenees with excellent food (tapas). It should be on everyone's itinerary.