Is there a train service from garucha to alicante

No. 

Is there bus transport from garucha to alicante or can anyone recommend transport

no there is very limited public transport in this area sorry.  the only option would be a private taxi which costs a lot or maybe the airport shuttle bus and then taxi from Alicante airport to Alicante.

I have seen a hell of a lot of alsa doing deals ads recently on Spanish TV,try looking at there web site and you might find something 

www.alsa.es.

its a good bus company that have been around for years and I know they have stop points in garrucha and vera

good luck

JJ

Like Jen-Jen pointed out ALSA are improving their service. But, an ALSA bus serving Mojacar Playa does not necessarily serve Garrucha and Vera.  And so it is with Garrucha wheir their service does not necessarily serve Vera or Mojacar Playa. But, if you can get an ALSA bus to Mojacar Playa or Vera the local bus service serves Garrucha pretty well. 

Bear in mind that if travelling from Alicante/Alicante airport you may have to change bus in Murcia. I've used the service Mojacar Playa - Alicante airport (changing bus at Murcia). It is good and cheap and you must book and reserve your seat online. The ALSA website could do with some updating. (from memory I think you must supply your passport number).

I wish!!  Only chance is  by bus.

If you look for a bus on the alsa website directly from Garrucha to Alicante, you'll find very little options. To get the biggest variety of times, search:

Murcia to Alicante, and put on a paper the times.
Vera to Murcia, and put on a paper the times
Garrucha to Vera by Autocares Baraza (not Alsa), and get the times.

Combine these three schedules as is convenient and buy three tickets.If you sign up on Alsa bus plus, you pay only once the administration fee. Also for future reservations.

There is an Alsa bus from Vera through to Alicante airport, stops at a couple of places on the way and change at Murcia. Takes just over 3 hours.

The alsa bus from Vera goes to Murcia bus station where you change for the Alicante airport bus. Also Exclusive airport shuttles do a daily run to Alicante airport. It is an early pick up from Garrucha and arrives Alicante about 9.00 am.