Well, having been born and brought up in that glorious island - I can offer some advice.
You will enjoy yourself. You will get wet, inside and out
If you start in Dublin, head out into the Wicklow mountains, OK, they're not really mountains by the standard of the Rockies, but plenty of hills to explore. Then, if you fancy something a little more relaxing, head to the coast and follow that down towards Wicklow town.
You will find that there's plenty of lovely places to visit inland, Kilkenny is well worth a trip, there's quite a spectacular castle there. On to Wexford and around the coast to Waterford, another pleasant town. Follow the coast to Youghal and Dungarvan. Oh! and there's a really nice beach at Clonea, just east of Dungarvan.
I also recommend a trip to Clonmel and a ride through the Comeraghs. I could go on, but you've probably got the picture by now.
A chap called Edward Enfield has written a very good book about a cycle tour around Ireland, you can probably get a copy from Amazon.