Traveling from Alicante airport to Murcia is pretty straight forward. For a couple of years now, the spanish bus company Alsa runs a bus service various times of the day to the city of Murcia. You can find details on the Alicante airport to Murcia bus here.
Alternatively, if you are traveling to Murcia only, you should check perhaps flying into Murcia's San Javier Airport (located about 50 km south of Murcia), which could get you quicker to Murcia and the LaManga area.
