Buy a one day Tourist Pass at any machine at an airport metro station (T1/2/3 or T4), currently 5.20 euro.
Take line 8 to Nuevos Ministerios (end of line).
Transfer at Nuevos Ministerios to Sol on Renfe Cercanias line. (Sol is one of the most important plazas in Spain.)
At south west corner of Sol (near the McDonald's) look for the Bus Turistic sign. Pay onboard (17 euros for one day, I think). There are two routes and you can hop on and off. A truly excellent way to see Madrid if you're a newcomer!
That's my recommendation!
As for the rest, buy a decent guidebook!