Finding timetables for the Alicante to Murcia services using the RENFE website is not immediately obvious as the page you come to at first only list long-distance trains, a small number of which do connect Alicante and Murcia, but in fact there is more to it than that.
You have to select Cercanias at the top right hand corner of the page and then Murcia/alicante.
This display trains that the previous page didn't display. The service is a bit more than hourly. The Murcia to Lorca service is also more or less hourly on weekdays and a bit less at weekends. Aguilas is served by three trains a day (from Murcia via Lorca) with one or two extra trains at the height of the summer.
The train fare is absolutely unbeatable: 5.30 Euros will take you all the way from Aguilas to Alicante.
Personally I am a fan of the Aguilas train as the descent at Jaravia provides some spectacular views of Aguilas, the coast and the sea. Seeing it is well patronised I'm surprised RENFE doesn't run it more often.
The connection to the airport C6 bus at Alicante is easier than it looks. The trick is that this bus runs in a loop, ie Alicante - San Gabriel - Torellano - Airport - San Gabriel - Alicante. This means getting from the airport to Torrellano is more straightforward that getting back. The easiest place to change is at San Gabriel as the bus stop here is directly adjoining the railway station and the connection works well in both directions.
The signposting at San Gabriel is not totally self-explanatory, but if you bear in mind that the platform on the beach-side is for Murcia and the platform on the land side is for Alicante, you can't go wrong.
If you want to change from train to bus at Torrellano, this is slightly more tricky but also doable. The bus does not serve the station directly, but is within easy walkable distance and is fairly self-explanatory.
I've never actually done any of the other airports by public transport so can't say how well they work.