Head north of Dublin, to the Royal County, Co. Meath. Just a few minutes up the N2 or N3 on the weekend, or an hour or so mid-week. It's home to Trim Castle, Hill of Tara, Kells, the Boyne Valley and River (home of the Battle of the Boyne- a fight between Irish Catholics led by James II a Scot and English Protestants led by a Dutchy, William of Orange) and the scared burial places of theBru na Boinne. Older than Stonehenge, Newgrange, Dowth and Knowth arethe burial places of ancient royalty.
Trim Castle is famous for being the movie set for Braveheart and its proximity to the Hill of Tara, a place at the heart of Irish folklore and legend, it was home of the high preiests and later a burial ground for high kings. Kells was the home of the book of Kells, until they moved it to Trinity College Dublin. You can see a replica of it better in Kells,without the hundreds of other tourists, but other than curhces and round towers there is not much else to see. Definately do a day trip to Bru na Boinne and do some reading before hand about the ancient history of the area as well as the not so ancient. The border counties were a hot bed of activity during the troubles, and while it is hard to get the locals talking about it, they have been going in for centuries now. Nearby Slane has a nice castle and a good hostel.
In the middle of all this great stuff is the raging metropolis of Navan. If you can avoid it, do, but otherwise there is a very funky chocolate shop with great coffee (rare in Ireland) in Metges Lane just off Market Square.