If you arrive in Shannon Airport and wish to go to the Aran Islands you will have two options. Option one is to hire a car and drive to Rossaveal through Galway. You can park the car at Rossaveal and take the boat to any of the three islands.
If you dont wish to hire a car there is a public bus service from Shannon Airport which will bring you to Galway City. When in Galway city you switch to another bus to Rossaveal which is provided by the ferry company and handily located just around the corner from the main bus station.