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Dublin
Things to Do  

In Dublin, definitely go see the Castle, see a play at the Abbey Theatre if you're interested in Irish theatre, the Book of Kells is in Trinity College Library, the old Post Office and similar places are neat to visit if you're into Irish history.

See a hurling match and a professional soccer match.

Day Trips  

Glendalough - just outside of Dublin, old abbey, very atmospheric

Powerscourt -  another trip out of Dublin, vast estate, gardens  

Howth - a little town on a peninsula just north of Dublin. There is an abbey in ruins there, and a good hike uphill to see a panoramic view of Dublin. This town is accessible by Dublin public transit.