Flying in and out of Dublin. Need to allot time for near Clifden and Milltown (near Tuam) in Galway for seeing where grandparents were from. What are recommended high points to fit in with such a limited time frame? What would you skip?
When will you be going?
Will you have a car or will you be relying on public transportation?
Is this your first trip to Ireland?
All else being equal, I'd use the RS guidebook to set priorities for Dublin. We were there for about 48 hours and enjoyed it very much, choosing just the sights that most appealed to us. We were extremely lucky to have mild, dry weather so walking tours were a lot more fun than they would be in more typical Irish weather.
Whether to head straight for Galway upon arrival is up to you -- you might (or not) want to spend your first 1 or 2 nights in Dublin to acclimate. In Galway area you'll need a rental car, yes? I'd probably take the train to Galway city and rent from there, but you can weigh the costs and the convenience.
Your last night before departure should definitely be in Dublin so as to avoid stress getting to the airport in case of any traffic jams or public transport delays.
First time. Anticipate driving when outside if Dublin. Second week of September.
You might consider taking the train (about 2.5 hrs., direct) from Dublin to Galway and renting your car there for the few days you'll have to explore Connemara. If you're arriving in Dublin after a long overnight flight you'll likely be pretty tired and might prefer to avoid getting behind the wheel until you've had a chance to get a good night's sleep under your belt before tackling the challenges of driving on the left.
Basing yourself in Galway would permit easy day trips south to the Burren and the Cliffs of Moher, to the west for a leisurely drive around the Connemara Loop via Clifden, the National Park for easy hikes, Kylemore Abbey and the village of Cong. You'd also be in a good position for your day trip up to Milltown. Depending upon your personal preference you could find a place in Galway, unpack once and settle in for a few days.
In addition you could drive up to the ferry landing at Rossaveal for the trip over to Inishmore. The Aran ferry company has a pickup point right in Galway (for a small fee) if you wanted to skip the drive that day. Suggest checking the weather the day before your trip to Inishmore to ensure that you'll have a reasonably sunny (and calm) day. Though the ferry will run in all weathers it wouldn't be much fun to be on the boat in rough seas. The ferry never fills up, especially that time of year, so you can wait to make a last minute decision.
Clifden would also be a good base, especially given its close proximity to the Sky Road - one of Ireland's great scenic drives.
Am assuming that you'll be spending a couple of your days in Dublin which means that your available time will be at a premium - all the more reason to make this easy on yourself and minimize the moving around by finding a single base and operating from there.