All you need is a rental car, a good map and family and friends to share your experiences on this suggested itinerary in Ireland.

From Shannon, in County Clare (point of arrival for most foreign visitors) consider the following scenic areas

Then drive to Ballinamore in County Leitrim. Take a quick detour to the ancient monastic site of Clonmacnoise with its beautiful grounds and gardens - what a great introduction to Ireland!
Spend a night or two in Ballinamore, a great place for a spot of golf, fishing and other outdoor activities. From here you can take a scenic drive to Newtongore and Leitrim.
Hit the road for Florence Court, located right outside of Enniskillen. This mid-18th century, Irish Palladian mansion is surrounded by gardens and rushing streams.  The house opens at 1pm for tours.

Another scenic route to take is around Lake Garadice on a road called Lakelands Drive.  The scenery is peaceful and, oh so beautiful.    
Next stop is Fenagh, a ruined late-medieval church of St.Caillin, located south of town.  Then on to beautiful Galway.

** Note:  The original information for this thread came from the following forum thread:

Please read the full trip report at the link provided by TA!