Hi Aida,
May is a lovely time to explore the Causeway Coast and Northern Ireland! 
Belfast has plenty to offer. You could consider visiting the 
Belfast Castle and the 
Titanic Belfast experience. 
Translink offers 
iLink Travelcard for rail and bus travel within Northern Ireland. You can choose the period and zones that suit your needs. The Causeway Coastal route is best done by bus. One of the best tools for planning your travel is Transport for Ireland's 
Journey Planner. More information about traveling by public transport is available 
here. 
Glenarm Castle would be on your way towards 
Rathlin Island and the 
Causeway Coast. Rathlin Island could be a great place to overnight as well. You could take a boat back to the Ballycastle in the morning and then a couple-kilometre bus ride to the 
Carrick-A-Rede Rope Bridge. You'd still have plenty of time to visit 
Giants Causeway that same day. Bushmills town and its famous brewery are close and could be a nice place to spend the night. There are also other castles nearby, such as  
Dunluce Castle. All the distances are short and bus services work generally well.
The whole coastal route is picturesque, with plenty of interesting places to see. Do you have some specific interest and how much time have you allocated for this part of your trip? Happy to answer any further questions you may have.
Cheers,
Lauri