What a pity you didn't inquire before I took the train to Murcia on Wednesday. Could have told you all the precise timings.
Train leaves Lorca San Diego 13 mins to the hour. Every hour during the weekday. So Alhama de M about quarter past the hour. Return journey train left Murcia @ 22 mins to the hour, during the weekday.
Unmanned stations have a ticket machine. Takes, cards and cash. You will need to buy an Ida and vuelta ticket.
Lots of taxis available at the station and there is a taxi rank outside the El Cortes Ingles on the LHS of main thoroughfare. Cost of taxi to El Cortes Ingles €7.30