Hi Billbo
Found this on another Forum.
"
Trains from Alhama to Murcia, the Murcia to Alicante (Main Terminus) don’t get off at small station by the sea.
Buy ticket Alhama to Alicante. Cheap 50% off over 65's
From Alicante station walk to Bus Station 500-600 meters from there there is a dedicated Bus to the Airport cheap.
Trains leave Alhama every hour at 11mins past the hour and return from Murcia at 15 mins to the hour (Don’t know Alicante times) beware last train from Murcia is about 9 ish (check)
Also if you want to go further From Alicante there is a Tram to Benidorm 3,85 € for a 1 hour 10 min journey over 45kms the tram is about 300 meters from Alicante station and at Benidorm walking distance of the old town."
http://www.tramalicante.es/page.php
Check renfe site for Train Times.Also there are Buses from Alicante to Benidorm.
http://costablanca.angloinfo.com/information/transport/public-transport/regional-bus-services/
Good Luck