My dear fellow traveller,
as someone who is half-Spanish and has studied in Madrid for half a year, I can only recommend you to see MADRID! Hala Madrid, as the Spanish hail the city
It is a great town but at the same time, I hear wonderful things from Barcelona where I have never been myself yet except for at the airport.
Also, wonderful is the coast from Malaga to Cadiz! Take a rental car and drive down there. I think Marbella, Estepona, Tarifa, Puerto de Santa Maria near Cadiz are some beautiful places to go to. Same holds true with Sevilla.
One of my favorite places in Madrid to stay is the CROWNE PLAZA CITY CENTRE. It is about 140 euros per night and located right in the middle of town. You get everywhere from there!
Let me know if you need more suggestions...
Martin
PS: Also take the time to watch a soccer game either in the Santiago Bernabeu in Madrid or in the Camp Nou in Barcelona