In Dublin check out
Trinity College - Book of Kells and Long Library
Grafton Street
Guinness Storehouse
Kilmainham Gaol
Christchurch
It's a very walkable compact city. There is also a hop-on-hop-off buss that goes to everything

Outside of Dublin we rented a car and headed West and North to check out Cliffs of Mohr and Giants Causeway. There is really too much to see in a week so don't try and see/do everything. Pick a few things to spend time on, mix with the locals and enjoy