Besides the Prado, what are the other sights/experiences I should plan for? We are hoping to go on the RS Spain tour which gives us a free afternoon and evening. We also fly out of Madrid so we could spend an extra day or two there before we return.
The Park Retiro is definitely at the top of the list...You can see much there, the people, the art museum, the crystal palace, the mimes and street performers, grab a bite of real Madrid 'fast food' and have a great experience.
The Old Postoffice, City Hall at the Cibeles fountain is also great, and nearby the Retiro. Get your ticket to the top as you enter on the lower flour and treat yourself to a magnificent view of the city.
A walk along Gran Via, now 100+ years of age is certainly doable.
And, if you have transportation, or can get it, check out the Pardo, Not the Prado art museum. Royal Palace, actually two of them side by side. Phillip II and Franco lived here as well. About 1/2 hour out from the city, but a great visit. And good food in the city at the sidewalk café as well.
I would suggest the Royal Palace and Almudena Cathedral, for sure, as well as the Plaza Mayor. I have no idea what the tour covers in Madrid, but the beautiful Retiro Park is right next to the Prado, and the Thyssen-Bournemisza museum is across the street and down a bit. You could visit the San Miguel market, walk down the Gran Via, or take a hop-on, hop-off bus for a look at both the old and newer parts of Madrid.
(HJ and I were obviously writing at the same time.)
Those are great tips. Let me add a more philosophical one:
Practice staying up late...
Yes, yes, yes to what Bill advised. Stay up well past midnight and you will find a venue of experinces so many tourist miss. Spain come awake after 10 p.m. and "alive" after midnight.
Very near the Prado, be sure to see Picasso's Guernica. Huge, in a room by itself at the Reina Sofia. Incredibly moving. Look it up to understand the history and the meaning.
I enjoyed visiting the Thyssen Museum (http://www.museothyssen.org/en/thyssen/home). I also enjoyed a guided tour of the Convent of the Royal Barefoot Nuns, aka Monasterios de las Descalzas Reales (http://www.patrimonionacional.es/real-sitio/monasterios/6095). You can also visit the Mercado de San Miguel.
Besides the Prado - There's the botanical gardens right next to the Prado that's pretty nice. Retiro Park is another good choice. I liked Temple Debod - but it has fairly limited hours so not sure if it will be open when you can go.
Prado for sure. Since you mentioned it, I assume you like art museums. So I'd suggest the Reina Sofia (modern art, Picasso's Guernica alone is worth the admission, but the whole museum is great). I have not been to the Thyssen museum, but apparently it's quite good, with a broad overview of centuries of art. Caixa Forum, near the Prado, has traveling exhibits, so it's worth looking to see if there's one you're interested in during your visit.
Retiro park is a great way to enjoy some nice weather - great urban park, gardens, fountains, and apparently the only statue devoted to the devil (fuente de angel caido).
The Royal Palace is another classic choice.
Personally, I like just wandering around near Plaza Mayor/Plaza de la Villa, Mercado San Miguel.