You will enjoy any of the destinations listed by others. Based on my one trip to the Southwest, I would consider your highest priority and choose based on that. Towards the end of the trip, we were pretty beat up by the wind, so you may want to consider an easier course.
For an easier course in a fantastic location with a relatively low price consider Dooks (35 miles from Shannon). 6000 yards, pretty easy course but interesting holes.
For the setting, I would consider Old Head. (100 miles) The golf course is a bit of a disappointment but it is a spactacular place. We finished our trip with 36 holes there on the final day of our trip and it was a memorable last day. Perhaps others can confirm or deny, but I recall hearing that the wind there is generally lighter than in other locations.
For the drive and challenge - Waterville. Waterville is a 135 mile drive from the Shannon airport, and for us it was the windiest course, but has some very memorable holes on the back 9 and was by far the toughest test we faced.
For a fun experience - Tralee. (85 miles) The back nine is outstanding and the front better than average.
We played Dromoland Castle and I was pretty disappointed (although the castle was cool). I also did not like Doonbeg (38 miles) based on one round, but would try it again.
I would double the drive time you would expect for the mileage in the States.