Just as you walk out of the airport you will find a long row of Taxis. The are used by most people on there first visit to La Marina.

123,
Make sure you ask the price first and also say, La Marina, San Fulgencio, because there is a La Marina near Benidorm, as a mate of mine found out to his cost.
Price can range from 30 to 50 euros, I think.

Taxi from the airport was €36 to La Marina Urb in July for 3 of us plus luggage. To do it by bus, you'd need to take the airport bus to Alicante and then bus from Alicante towards Cartagena I think, getting off in the village, then taxi or walk (haha) from the village to the Urb.

I'd just take the taxi from the airport...

There is a bus every hour (daytiime) from the Airport to Elche, timetable can be found on line. We often get this bus and then get the bus from Elche to the Urb. It costs about Euro 7.50 for the two of us for both buses. You can also pick up a bus from Alicante Bus Station direct to the Urb - buses to Alicante Bus Station from the Airport are every 20 mins. It all depends what time you land which is your best bus option (and if weekend / red day etc) but if you're landing in the evening, bus is not a great option.

We always use Suntransfers.com The cost is €74 return for up to 4 people. Sounds pricey, compared to the bus options, but they're always waiting for you at the airport when you arrive and bang on time to take you back again. You also pay in advance when you book online so there's no disputes over the cost. I highly recommend them, especially on your first trip when you don't know the area.