Does the bus or subway go to the closest train station to the airport? which train station should I use to go to Mucia?

Let's assume you will be arriving to Barajas airport in the morning and you want to take a train to Murcia, southest trains leave from Atocha train station.

My option, depending on how heavy your bagage is, will be taking the subway from the airport (airport station line 8) to Atocha Renfe train station where you can get your train to the southwest.

The subway is all you need for the whole trip

http://www.metromadrid.es/ look on Plano de Metro

Apart of the option Nativo said (i.e. metro all the way from the Airport to Atocha Renfe, transferring lines two times), there's another one, a bit more comfortable if you are carrying luggage.

That is, taking the metro (line 8) from the Airport to Nuevos Ministerios (the end of the line) and there transfer to the commuter trains system called Cercanias. It actually works as a metro inside Madrid with very high frequency of trains.

In the metro station, look for the sign of Cercanias (kind of a white rotated C on a red background) and there go to platform 4 (IIRC, there it stop the southbound trains to Atocha). Anyway, it is well indicated. Once you arrive at Atocha-Cercanias, follow the signs to Grandes Lineas (Long distance lines) or AVE. It will take you to the adjacent Puerta de Atocha station, where your train to Murcia departs from.

You will pay an additional euro to use the Cercanias, rather than only metro, but you will get to Puerta de Atocha faster (trip from N. Ministerios to Atocha-Cercanias is about 7 minutes long, and I'd be very surprised if you have to wait more than 5-10 minutes for the trains) and in a more comfortable way.

Have a nice trip.