Not very much public transportation available. If you are going to Alicante, the airport bus run quite frequent, but for other destinations you have to plan ahead for the bus timing.
For Calpe or Benidorm, use Alsa bus, that run every now and then. Buy online (or in app), but online sales close 30 min before departure. It is also possible to buy ticket by the bus, but then it is cash only.
https://www.alsa.com/en/web/bus/home
I was going to Calpe but just missed the direct bus, so went to Benidorm to change there. Price was 11.10Euro from the airport to Calpe, a prebooked taxi would be around 120Euros.
For other destinations, check
https://www.alcairport.com/bus-travel-alicante-airport/