What a lovely month for your trip! First take the Bus ride, called Madrid Vision. It gives you an overview of the center of the city. While you did not say how long you would be in Madrid, I would suggest that you take at least one day for the Museo del Prado, anything less is barbarism!
Madrid is not just a place it is an experience. Enjoy strolling up and the the Calle de Alcala and La Gran Via! Chill at the Puerta Del Sol!
Have a nice desayuno at one of the little bares and walk to the Plaza Mayor, be sure to stop in at the wonderful bakery along the way,La Mayorquina, and have some Palmeritas!
Past the Plaza northwest onwards to the Palacio Real and take the tour. On the way back go to the Teatro Real near the Plaza de la Opera, lots of fun there.
Spend a nite at the Plaza de Santa Ana in front of the Hotel Reina Victoria, its a lively place and you will see Little Lorca, a small statute!
Shop till you drop at the Calle Preciados, and of course the Corte Ingles!
I would recommend a day trip to Segovia and El Escorial. If you have the time another day trip to Toledo.
Buen Viaje,
Carmen