If you are coming to the Aran Islands from Belfast either by car or public transport, it is advisable to go first to Dublin and then cut accross east to west to Galway.
Public transport services between Belfast and Dublin and Dublin and Galway are very frequent and affordable. If you are coming by the car the motorway is excellent. You take the M1 highway route from Belfast to Dublin and the M4 and 6 to Galway. Then you have to drive out to Rossaveal which is another 1 hour. If you do not have a car there is a connecting from Galway with the Ferry company.