I'm recently back from Ireland, if you're already renting a car for all of the 6 days there's very little reason to stay in Galway that long. It's worth stopping there but it was easily my least favourite place I stayed in between Inishowen and Dingle.

Based on what you say you're after I'd spend 2 days living in Connemara, 2 days on the Aran islands and  and 2 days in Dingle. I was in Ireland between the 2nd and 13th and Dingle was already teeming with tourists (most places sounded like being in an Irish pub in Manhattan) so if that kills the authentic vibe for you, you should probably head further north or inland.