First post of flyertalk so be nice!

I'm from Northern Ireland originally and there is no way I would attempt to do Gorey->Giant's Causeway by public transport. The public transport is poor at the best of times in Northern Ireland.

You would need to get a train from Dublin to Belfast, and then probably change trains (and stations, I think) to get a train to Portrush. You can probably then get a bus to the Giant's Causeway.

However, the bus services are better than the trains, in my experience so you would be better off changing to a bus in Belfast. I believe there is a direct link from Belfast to the Giant's Causeway, but it will probably take a couple of hours.

http://www.translink.co.uk/

is the public transport website for NI.