I am a regular visitor and, while I still have a lot of ground to cover, can offer a few ideas.
Dublin – head south to the Dublin/Wicklow mountains where popular highlights include the Sally and Wicklow Gaps. The climb from Laragh through the Glenmacnass Valley to the Sally Gap is particularly good. This
article from Cyclist.co.uk should whet your appetite.
South East – consider the new
Waterford Greenway, an off-road route along the line of a disused railway from Waterford to Dungarvan, then add on all or part of the
Sean Kelly Heritage Route to explore the Blackwater Valley.
South West – the
southern peninsulas of Dingle, Iveragh (Ring of Kerry) and Beara are beautiful and lend themselves very well to touring loops. Killarney is a good base for all three or Kenmare for Iveragh and Beara.
The Ring of Kerry is a popular (and busy) loop. Go clockwise so as not to be overtaken by tour buses which by convention go anti-clockwise. Add on the much quieter “Skellig Ring” and Valentia Island at the western tip. The interior Ballaghbeama Gap is off the main tourist trail but makes a good ride between Waterville and Moll’s Gap (though it would a shame to miss the classic climb of Moll’s Gap from Kenmare).
I love the Beara Peninsula which is quieter and hillier with some good climbs including the Healy Pass. The Dingle Peninsula is still on my to do list and includes the famous Connor Pass (highest in Ireland I think).
West – the Burren is a fabulous area for cycle touring (go in May/June for the wild flowers). Friends and family who have done the “Tour de Burren” sportive said it was a great route. It is not shown on the event web site but should be easy enough to find.
The
Great Western Greenway from Westport to Achill Island is another off-road route along a disused railway, it’s only about 40km IIRC but could be included in a tour in that area. Achill is spectacular.
Also, ask the locals: