If you want to use subway (metro in Spanish) you can go to the nearest Plaza de España's station and there you can take the line 10 direction Hospital Infanta Sofia and get off in the 5th stop that is Santiago Bernabeu station, that also is the name of the stadium. Only one line, no changes, straight to the 5th station. An other option is to buy a excursion or tour, from the turist office inside the Madrid airport and I believe they will pick you from your hotel and include a tour inside the stadium. If you prefer to go on your own because is very easy and cheaper you can buy a Stadium tour on the stadium's ticket office itself.
It is NOT near, but you can use the metro to get around the whole city. You also ac buy a turist ticket for the amount of days you are going to be there, in Madrid airport there is a tourist office where you can get your tickets and some maps, also you can buy there some tours to another cities near Madrid. Toledo is a MUST!