Madrid is a beautiful city full of life and culture. I got a previous request to do Madrid in 48 hours. Even though 48 hours does not seem like a lot I know you can conquer the city in that amount of time. My suggestion:
Day One: Hit all the majors
For Day One I would get as much of Madrid as I could. I would suggest visiting Sol, Gran Via, and Plaza Mayor. They are all in the same area so going from each will not be hard at all.
After seeing the wonderful characters at Plaza Mayor (and yes I mean Mickey Mouse & "Fat Spider Man") I would make my way to San Miguel Marketplace where you can get a variety of tapas like croquettes, sushi, wines, fish, etc. Sit in the middle of the marketplace and enjoy the Spanish ambiance.
With your stomach full next stop is the Royal Palace. It is simply gorgeous.
After that I would rest my feet, enjoy a nice (but small) siesta and prepare myself for Madrids nightlife. My suggestions: The Cave Bar, Areia; a bar with beds and huge cocktail list, or a night in Chueca (Spain's homosexual neighborhood- It is a blast.)
Attire: With all the walking, sight-seeing, and culture you are about to encounter my attire would be. A cute spring.summer day dress and a pair of your most comfortable flats. For night, jeans and a tank-top with sky-high heels.
Museums; Prado, Thyssen & Reina Sofia
The Prado and Thyssen museums are close to each other but the Reina Sofia is a little farther so before hopping on the metro (in the spring/summer I prefer the bus) stop into Retiro Park. Pack yourselves a lunch (a lot of restaurants have picnics to go) or snacks and find the best piece of grass and enjoy a picnic. Before leaving the area make sure to stop by Banco de Espana (now known as the post office) it is simply gorgeous. After leaving Retiro head to the high-end shops on Serrano Street, I would go even if you just window shop. The shops themselves are beautiful and there are well-known designers to Spanish designers like My favorite; Agatha Ruiz de la Prada.
Flamenco at Corral de la Moreria; i would arrive early for dinner & the show. they have a wall filled of photos with each celebrity that has visit this Flamenco place who knows who will be sitting next to you! (Jenifer Aniston was there just last week)
Attire: This day is a little more casual so I would stick to jeans and shorts with converses and a short-sleeved tee-shirt or tank top. Everyone at any age wears converse shoes so you will have no problem not looking like a tourists! But, be careful of pick-pockers! They'll get you.
Museums museums museums. On a rainy day (hopefully you will not encounter this) the perfect place to be is at the museums with all the famous artists like Goya and Botticelli. I would also check out Instituto Cervantes (calle Alcalá, 49) they always have new and interesting expositions.