Burgos has an airport with connections to Barcelona. Check the flight schedules at www.iberia.es. Burgos airport is connected with the city center by public bus and taxi.
The most convenient option is to arrive at Madrid Barajas airport and, from there take a direct bus to Burgos with Alsa company from Terminal T4. Or public transport («Cercanias train») from T4 to Madrid Chamartin (20 min) and then link with regular Renfe train to Burgos.
Other close airports are Valladolid, Bilbao and Santander.
Alsa bus company has frequent bus connections with Burgos. The Burgos bus station is located in the city center.
In Madrid, Alsa bus station is located in the transportation node «Avenida América». The bus trip Madrid-Burgos takes around 2h 30m.
To arrive from Madrid Barajas airport there is a direct bus from Terminal T4 to Burgos with Alsa company. You can see the schedules in this link.
Burgos train station “Rosa de Lima” (www.renfe.es) is 5km away from the city center, but there are buses and taxis to get to Burgos. Connections with Barcelona, Bilbao, Madrid and other national international cities are available.
From Madrid Chamartín station there are around five daily train connections to Burgos.
Finally, from the train station in Burgos you have to take either a bus or a taxi in order to reach the city center.