The first thing that caught my eye when I looked at a map of Ireland was the highest point in the country (of course) - Mt. Carrantuohill.
This was our first planned activity in Killarney. It's about a half hour ride from city center by cab. Make sure you tell a bunch of people you are going and that you bring hiking supplies (water, food, warm clothes for the summit). Going up and coming back down should take a total of about 4 hours without dilly-dallying.
In our haste to climb Carrantuohill (part of the Magillicuddy's Reeks), we later learned that we actually climbed a mountain next to it. It might've been something like Cruagh Moher, but I'm not sure. This could've been a good thing, as our host at the bed and breakfast said that experienced climbers (which we most certainly are not) have gotten injured and lost recently while up on Carrantuohill.
One more tip on Carrantuohill: try to go on a day when it hasn't been raining for a couple days (hard to do in Ireland). It was kinda tough even to get to the base when we went since it was raining pretty hard and we had to cross mud flats. Then again, I've read that Carrantuohill actually has trails to the summit, whereas whatever we climbed did not.
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