Madrid is a great City, go to the Las Ventas bullring, very impressive, did the tour and was blown away and learned a fair bit.
Head to the Plaza Mayor, which is a big square and a nice place to have a drink in the sun, San Miguel Market is a great place for some proper local tapas where you will find all the locals eating and drinking. And of course make sure you go to the Bernabeu and do the tour and watch the game if they are home.
And as for getting around use the Metro buy a 3 day ticket very cheap and easy to get around the City. Enjoy