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.
Pamplona is best-known for its annual "running with the bulls"... Yes, these are real bulls and it is dangerous! Only the very sporty ones that have something to prove, should try this...
San Sebastian is a superb beach location just west of the Pyrenees with excellent food (tapas). It should be on everyone's itinerary.
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.