Either way you can't go wrong.
Galway is a very lively small city, easy to navigate with lots of nightlife and trad in the pubs. There's also an appealing pedestrian-only area smack in the middle of town.
Doolin, on the other hand, is the mecca for traditional Irish music, if that's of interest, with O'Connor's Pub and two or three others an easy walk from each other so you can sample a bit of each.
Coming from Dingle, the Cliffs of Moher are right on the way to Doolin, which gives you the option of spending more time at the cliffs and cutting drive time for the day before heading to Connemara.
If you opt for Doolin, I recommend the Fisherman's Rest B&B, which is right behind O'Connor's. Danny and Mairéad are superb hosts (heck, I can say that for just about any Irish B&B in which I've stayed!). The website is:
www.accommodationdoolin.com/