From Sligo to the Aran Islands, if you are taking public transport you would need to get a bus to Galway City which is around 1.5 hours. In Galway City you would need to switch and get a bus to Rossaveal. The ferry companies have an inexpensive bus service to Rossaveal.
If you are going by from Sligo it would be advisable to take the scenic route and head towards Castlebar and Westport. This direction will take you down through Connemara which has incrediable scenery. However it is advisable to use Sat-Nav in the as it is easy to take the wrong turn in Connemara . The driving time from Sligo through Connemara and Rossaveal is about 1.5 hours.