Hi Aida,
Exploring the Causeway Coast without a car is entirely feasible, as mentioned by Lauri, the Causeway Coastal Route is best explored by bus.
From Belfast, I would suggest you take the train to Coleraine Bus/Train Station. From here you can pick up the Ulsterbus 402 (Causeway Ramler) Service to hop between the major tourist attractions.
A suggested itinerary might be to visit the Giant's Causeway first (the recommended time here is 1-2 hours), then hop back on the 402 bus to head to Carrick-a-Rede Rope Bridge (the recommended time here is 1.5 hours and pre booking is essential). From the rope bridge you can then take the 402 service again back towards Bushmills/Portrush and get off at Dunluce Castle (suggested time here is 45 minutes). After a full day of sightseeing, hop back on the 402 bus back to Coleraine Bus/Train station where you can pick up a train back to Belfast.
Hopefully you find this information useful!
Best Wishes
Alison