From the airport
Metro: L8 to Nuevos Ministerios then metro L6 to Diego de León. It'a a 46 minutes trip. If you are in the city for a few days buy a T-10, Multi-person card for 10 journeys, (€12,20). It works on the bus too.
Taxi: It's a 20-30 minutes trip and is around €30.
Aerobus shuttle: Drops off at Atocha, Cibeles (night) and O'Donell-Doctor Esquerdo to Airport T1,T2 and T4 and costs 5€. Bus: Line 200 to Av America then metro L7 to Parque de las Avenidas stop. It's a 53 minutes trip.