Take your time to explore Country Clare.
The Cliffs of Moher are very touristic with lots of people, the not so famous cliffs (south) of Killrush are also nice to visit on a rainy or stormy day!
Try visiting the Aran Island (in particular Inishmore). Go by boat (from Doolin) and rent a bike to explore the islands.
Go walking in the Burren or drive through with your car. Take a few moments to enjoy the mistic Poulnabrone Dolmen.

If you want to go for a city break, Dublin is fine, but 2 days are enough to visit almost everything! The city is not huge. I enjoyed going for a pint in Temple bar (Quays Bar), with Irish music almost everyday!

Guess what - you can meet ghosts in Knock, and I'm not lying. I met mine 8 miles north of the town, near the turn to Connemara. He was an old guy with a handlebar moustache, a grey ducktail and green socks, riding an old bike.. it's a good story, but too long for here! Go to Knock and get your own personal ghost story.

Go walk around the old castles and abbeys that are everywhere in the surrounding small towns.
Not much else to do in the tiny towns but there sure is plenty of green.
Also, eat a "99" from any gas station or shop, it's ice cream with a chocolate stick in it and it's delicious!

dublin is a very old but pretty city and quite large

If you ever go to Dublin, don't miss to visit the Guinness storehouse! It's a fantastic museum located in the original Guinness factory and gives an excellent insight into the beer brewing process, which is interesting even for people who don't drink beer (such as me for instance). Have a look at the souvenir shop (great things for your friends and family to take home, but a strain on the budget...), stroll through the several floors (especially the ads are really worth paying a visit) and don't forget to go to the bar at the top of the building, where you enjoy the view and a drink for free (e.g. a pint of guinness after which you won't need to worry about dinner because you won't be hungry anymore...). If you fancy a typical Irish dinner, go to "TheShack", a restaurant in Temple Bar (very trendy area of the city with plenty of bars, restaurants etc.) and have a traditional Irish Stew. But there is much more to see and do in Dublin!

Go to killiney with the dart, and don't forget howth! Search for the bay of Napels in Ireland!

Great fun, perfect live music at the pubs...

Flew to the Shannon airport and went to King Johns castlein Limmerick. Then Cliffs of Moher and Blarney Stone castle in Cork. Nice castle and lots of history, but other sites might warrent your time.

Look for the harristown portal tomb, its very impressive, the rocks contain lots of quartz, in a typical location over looking a valley with a stream running right beside it. There is also a standing stone in the next firld.