When Saint Patrick introduced Christianity to the Irish, he often founded a monastery to keep the flame alive. And from 432 to the dissolution of the monasteries under Henry VIII monasticism flourished in Ireland. First in a specific "Celtic" way, later spearheaded by the European orders. Ruins and remains of monasteries still are numerous in Ireland - and I encourage you to include a few in your plans.
Glendalough - County Wicklow
Here Saint Kevin founded his monastery. The solitude in the Wicklow mountains must certainly have appealed to the monks turning away from "worldly life". Even today the way there is not the easiest. And though the monks have long left, Glendalough's impressive remains (including a cathedral and a complete round tower) tell of past glory.
Monasterboice - County Louth
Nendrum - County Down
This was a "lost" monastery and its location on a remote island in Strangford Lough makes it all the better for it. Though its round tower is but a stump and other remains are sparse, the small visitor center tells the interesting story of this settlement. And on a good day the view from Nendrum across the lough is simply stunning. Be prepared for a bit of a curvy drive, though.
Kells - County Meath
Though the modern town has encroached upon it, the monastic district formerly found atells is still identifiable by the road layout. Which does not endear Kells to the motorist. The round tower in a corner of the churchyard contrasts with the later, partial medieval church spire. And a number of high crosses can also be found - one in an interesting incomplete state.
Mellifont - County
Just a short distance from Monasterboice, Mellifont heralded the advent of "Continental" monasticism in Ireland. The buildings were laid out to exacting plans and most csn still be traced today. Though Mellifont consists largely of ruins, the splendid lavabo bears ample witness to its past glories.
Fore Abbey - County Westmeath
You might be mistaken at first - from a distance Fore Abbey has a certain "castle" feeling about it. Not without reason, as this was a fortified monastery built to withstand the odd unfriendly visitation by less pious contemporaries. Even in its ruined state it still conveys a sense of power and security. The best views can be caught from the raised dovecote.
Bective Abbey - County Meath
Another monastery resembling a fortress at first glance, Bective Abbey seems to guard the Boyne crossing nearby. Many parts of the building are still fairly intact, though the cellars are not readily accessible. A place to explore ... once you find the entrance.
St. Mary's Chapterhouse - Dublin
This is one of the hidden attractions of Dublin - literally, as the chapterhouse of once mighty St. Mary's Abbey (which gave Abbey Street its name) is underground today. And incorporated into later buildings. Seldom visited by tourists it is a rare treat. Though the building itself is simple, its history is fascinating. And it'll give you a Dublin memory not many other visitors share.