spain

With that short of a time period, you should stay in Madrid for easy transportation options. Royal Palace in Madrid is a must-see and if you like art, The Prado. If you want to see the aqueduct, you could take an easy day trip to Segovia on the train from Madrid.

You have to get a guide book where you will discover your own "must sees" based on your interests and so you get an idea of the distances between where you want to go by looking at a map of Spain. You don't say what city you are flying or taking the train into

IMO I think Barcelona is more interesting than Madrid. More to see. 2 full days in Madrid and the rest in Barcelona. I agree with the other post. Google things to see in Madrid and Barcelona and pick what interests you.

With only 6 nights, 2 cities is plenty. I would stay 3 nights each in Barcelona and Madrid. The white villages are too far and travel to them would take up too much of your short time. You could take an easy day trip (30 minutes by public transportation) from Madrid to Toledo to see the contrast of the old Columbus-era capitol with the more modern Madrid. Toledo also has damascene artisans, Mudéjar architecture, and a great ornate cathedral, among other interesting sights. Be sure to check the time for the return to Madrid; I believe the last return is in the late afternoon/early evening.
In Madrid, I second both the Prado museum (there are two other very good art museums, Thyssen-Bornemisza and Reina Sofia, nearby as well) and the Palacio Real. The relaxing and interesting Retiro park is also close to the Prado. A bit further to the north is the Museo Arqueológico Nacional (archeological museum). If you like churches with great artistic interiors, the San Francisco el Grande is a few blocks south of the Palacio Real on Calle de Bailen.

I am flying from Austria and from Madrid and Barcelona flying back home to Canada