The most practical ferry route is from Rossaveal, about 40 km west of Galway city, with a crossing of roughly 30 minutes. Aran Island Ferries runs this service and coordinates with a bus from Galway city centre — book both together on their website, especially in summer when the boats fill up. The Doolin ferry from County Clare takes about 90 minutes and is more scenic, but runs less frequently and is more weather-dependent.
Aer Aran Islands operates small plane flights from Connemara Regional Airport near Inverin. The flight takes about seven minutes, which is genuinely absurd, and the view of the limestone pavement from the air is spectacular. It is more expensive than the ferry but worth considering if you are short on time or the sea is rough.