From Dublin: You can take the Enterprise train from Dublin (Connolly Station) to Belfast (Great Victoria Street), Belfast is also served hourly by coaches from Dublin and Dublin Airport. You can also take a Bus from Dublin (Busáras) to Derry which is a journey time of four hours. From here you have the option of train or bus along the coast to Coleraine and the Giant's Causeway.
From Belfast: For a summer time visit (28th March to 28th September) you can catch the Antrim Coaster which runs a service up the coast to Coleraine via the Giant's Causeway. You can find details of this here.
BY CYCLE: It is possible to hire bikes in Belfast and other locations and follow cycle routes: www.cycleni.com
check out the cycle routes here.
BY AIR: Belfast International Airport, George Best City Airport, Derry City Airport