Santiago de Compostela is connected via fast train to Madrid, see www.renfe.com
From the airport of Santiago de Compostela (SCQ) to the city center, you may take a bus or taxi.
There are two types of buses: a fast one going directly to the train station, and a slower one making all stops until the city center (in the second case, get off at "Plaza de Galicia").
For information on bus from/to the airport visit https://www.tussa.org/web/linas.php?id=41&lg=cas and http://www.tussa.org/mediateca/Horarios_TteUrbano/AEROPORTO_Tte.Urbano_L6A_e_20240909_dtp.pdf.
A bus ride costs 1 EUR. Tickets may be bought on the bus only with cash (and not with big notes). Information on buses may be found also in the apps "Máisbus" (also in English) and "Bus Santiago" (only in Spanish).
One taxi ride from/to the airport lasts around 15 minutes and costs 23 EUR.
Other nearby airports are in A Coruña and Vigo.
Santiago de Compostela is also the final point of the famous pilgrimage walk.