Hello,
Major things to do in Sevilla:
-The Cathedral, 3rd biggest in the world. Amazing place.
-The Alcazar is spectacular. Book a few days in advance to avoid waiting lines
-See a Flamenco show. We saw 2 in our 2 trips, we loved Casa de la Memoria. Small and authentic Flamenco. Also good to reserve a day or 2 in advance
-Plaza de Toros. You won't feel at home which is great. We decided to just take the guided tour with museum. My friend actually went to a bullfight. They loved the atmosphere.
-Plaza de Espagna
-Maria Luisa park just in front.
-Eat tapas, drink sangria or Vermouth. :-)
-Just walk the small streets of Santa Cruz, it's gorgeous
-Metropol Parasol for some modern funny and different architecture.
-5-10 minutes from there, a very nice museum: The fine arts museum. Peacefull, not crowded
-Casa de Pilatos, some great Mudejar architecture
-If you want to do an easy day trip by train, go to Cordoba. 45 minutes and you are there. The Mezquita is fantastic, the jwish neiborhood as well
Madrid
-The Royal Palace
-The plaza Major and surroundings
-La Latina disctrict(great for tapas bars and restaurants
-San Miguel Market
-Puerta del Sol
-3 major museums: Prado, Reina Sofia and Thyssen
-Retiro park in the same area(with christal palace and lake
-Plaza de Cibeles
-Puerta del Alcala
-Day trips could be, the magnificient town of Toledo, The El escurial Palace, the town of Segovia with the great Roman Aqueduc.