It's kind of touristic, but the Ring of Kerry is gorgeous. The Ring of Skellig (an extension of RoK) is not as common. Not sure how you'd get out there without your own wheels...
Skip the Cliffs of Moher, (although Doolin, which is nice, near the Cliffs seems to cater to backpackers) skip the Dolmen, spend the least amount of time in Dublin (not represent of Ireland at all). Waterford is cute, port town-ish though.
If you're feeling bold, hitchhike! I haven't, but know people who have, and I have heaps of family there, and have met enough random locals that I'd feel comfortable enough doing it, and I'm not a huge risk taker.
For some random Irish history, Cong, County Mayo is The Quiet Man was filmed. It's an old Irish movie with John Wayne and Maureen OHara in it. Anyways, there's a lot of stuff dedicated to the movie there. I was there a few years ago with my family and the owner of the pub the cast hung out in closed the pub but let us stay as long as we like!
I know this bit of information isn't very helpful, but at least you know it can be done: I met a young couple hiking Ireland a few years ago. They had a map of hiking paths (which I guess run all over the country) and their tents and raingear. Don't forget the raingear!!
And about the weather: there's never bad weather, only bad attitudes and wrong clothing. Or something like that
