There is a special bus line that connects Barajas Airport wih Madrid city centre, more specifically with the Avenida de América and Plaza de Colón. Buses leave aproximately every 15 minutes.Tickets cost around 2,4 €.
If you wish to continue your trip to Burgos, you will need to get off at Avenida de América, where you can take a bus of the company CONTINENTAL AUTO S.A. If you wish to stay in Madrid city center you should get off at Plaza de Colón (final stop). It is a 15-minute walk to El Retiro Park and El Prado Museum.
For further information about Madrid, please, follow this link: