Hi Scapa....
You can take the train from Alham to Murcia then change to take a train to Alicante..... depending on the connection time (which depends on the day of the week) you would be looking at 35 ish minutes from Alhama to Murcia & then 1 hour 20 from Murcia to Alicante......
Check out http://www.renfe.com/ then on the left hand side go to Cercanias and choose Aliante/Murcia
Another option is the bus direct from Alhama to Alicante. Check out http://www.alsa.es/portal/site/Alsa/
Good luck
Kathy ex-Dragontours... now in the MAPFRE office on Camposol B