A little bit south of Dublin is an old monk site called Glendalough, which was pretty cool too.
We went there on our first trip and my wife started walking around the cemetery and half of the stones had her maiden name on them. Turns out that's her heritage - family originated from County Wicklow. We had no idea. Was pretty darn cool.
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So I've been twice. If you're starting in Dublin I recommend just getting out of town and hitting the countryside. Rent a car (with the wifi hotspot!). Pre-rent as they insist on you getting the super cover (Doolin rental cars is what we used). But, really,
you need a car. The best stuff is off on small roads. So you get a great adventure to start - driving through Dublin on the wrong side of the road with a stick, shifting with your left hand (automatics were about double to rent).
Oh, yeah - good times. We did a counterclockwise loop from Dublin. IMO, try to see both historical and culture items - we went to a hurling game in Thurles and had an absolute blast (good suggestion above, there!). Things you should try and see:
Newgrange - Just unreal. Really, this is a once-in-a-lifetime site to see. Pre-book, as it can fill up quick.
Galway - used this as a hub for a few items. But inside Galway definitely, definitely see
Trad on the Prom. Awesome music and dance. This used to be the group The Chieftans, who have now grown up, gotten married, had kids. The music is incredible and the dancers are the best in the country.
Connemarra - as long as the day is nice, do the five mile loop up the mountain. Once of the most dramatic, prettiest place on earth. (Note - this is a decent hike as the last bit goes well up!). Pair the day with
Kylemore Castle.
Aran Islands - ferry is close to Galway. Some great neolithic sites. And definitely answers the question "What do you do when the entire island is covered in loose rock?"
We biked around the island - great times. I'd put Connemarra above this one, though.
South of Galway - Either as part of staying in Galway or making your way to Killarney (yes, you need to go to Killarney!). Definitely
Cliffs of Moher. Way cool. Also hit up
Paulnabrone, if you can
. Nice if you have the time and you can say you've seen a dolmen. 'Cause these things are important! If you're the hiking type I also recommend a hike in
The Burren. Absolutely unique land - takes a bit to get there and find your way. I have a picture of my kids with that as a backdrop that will stay in my house forever. Just stunning place.
Then get yourself to Killarney. The Park is amazing. Definitely find your way to the
Gap of Dunloe. I biked it (twice). Lifetime memory. Easily top 5 for most beautiful place on earth.
Waterford was definitely my favorite city there. Incredible history (oldest city in Ireland), great museums. Historical museums from viking days on. Waterford crystal museum was awesome. I recommend a full day there. Loved that place. It's on the way back to Dublin, so makes a great last stop.
Blarney Castle was ok. Liked the grounds and the poisonous plant garden more than the castle. The dinner they put on is pretty cool. That said I liked
Cahir Castle a lot better - neat place. ETA: If memory serves they allowed the kids to climb all over the walls here. Went over well with mine.
Also, super important! Dublin and Shannon are two of the very few places where you clear customs there, and not in the US.
Make sure you have plenty of time before your flight to clear. I'd recommend 3 hours, at least.
I made a travel packet for myself on this trip, so if you're interested in this kind of itinerary I can share it.