On our first trip to Ireland, we started in Dublin (of course), then traveled west stopping at Kilkenny, Irish Stud Farm, and spending a lot of time on the Dingle Peninsula. Driving the Peninsula was a highlight, and I highly recommend staying at least two nights there.

My surprise favorite thing was a trip to the northwest where Burren with its otherworldly rock formations.
Westport was also a lovely little town with some cheap and gorgeous accommodations. (Always order the full Irish).

I just went back a few weeks ago and this time we traveled north to south - again totally scenic. Northern Ireland was legendary with the Causeway Coastal Route well worth the trip. It is full of fantastic sites from castles to the Giant's Causeway. We ended down south in Wexford which was fairly quiet, but had a charming main street, great pubs, and an amazing beach.