Galway or Dublin ?

Flip a coin? Whatever works better logistically? Galway itself is small, so probably a better one-night destination. But just a glimpse of Dublin is not terrible if it works out better.

That's a tough one. From your other post it appears that you haven't been to either city before and you don't plan on renting a car. If you are leaving Ireland via Dublin you might want to save your last night in Ireland for Dublin. To maximize time on the ground you'd take the bus to Galway directly from Dublin airport. But if you're leaving via Shannon (or Kerry Airport) then you might want to spend a day in Dublin - that's assuming you'd be awake enough after an overnight flight to actually enjoy Dublin.

A little context would help. Who is "we"? What are your interests? What time of year are you traveling? I see from the other comment that you are not renting a car, so how are you getting to Doolin? Seems like that would be easier from Galway. Where else are you going?
If you decide on Dublin, what will you do with your bags until you are able to check in to your accommodation? There is a left luggage at Trinity College. One option would be to go into Dublin city center, leave your bags at Trinity College and walk around and see the sights (a good way to start any overseas trip), then take a late afternoon or evening train to Galway. Doesn't give you much opportunity to see Galway, but it gets you closer to the "real" start of your trip.

Thanks all- decided to stay the first night in Dublin. I forgot- jet lag is even more real the older I get!