You can do quite a bit in Madrid. They have a world class museum called El Prado which has some really fine art, you can tour Palaces, they have beautiful parks and squares, one of the best soccer...
around 25km, but it is very well connected by car, via the first ring road (M30) northbound, taking the exit to A2 highway and then going straight ahead for 10km. A taxi trip is around 25€ off-peak...