Hi Elizabeth,
Shame to stay only 5-6 days as there are so many beautiful places to visit.
I'm assuming you .flying into Dublin? If so, I would say stay one night/one day in Dublin. Visits not be be missed in Dublin, Guiness Brewery...it's amazing!! Night life in Dublin: Temple Bar is the place to be. If you are still alive the next day, and not too hungover and feel like driving, go right accross the country to Galway. This a vibrant, fun city. Normally motorway all the way there but count 3-5 hour drive. If you don't mind driving, from Galway, go south to Clare and the Cliffs of Moher. This is the most spectacular place in Ireland. Driving from Galway, go along the Coast you will visit fabulous little villages like Dullin and Lisdoonvarna, and the scenery along this coast is awesome. If you get that far, drive to the peer in Dullin and walk along the rocks here. If the sea is rough, the waves crashing along these rocks are breathtaking! From here, you can take a boat out to the Aran Islands. Don't go to the bigger Inismor Island, but rather one of the smaller ones. On the Island, you can take a horse driven trap around the island and stroll on the beach and listen to the local brogue. Everyone speaks Gaelic on the Islands!!
Now, if you prefer not to drive too far, and believe me if you take the above trip, you will spend a lo t of time in the car, you could check out the East Coast - to the South, the Wicklow Mountains, notably Glendalough, it a beautiful magical place. North of Dublin you could visit historial sites of Slane Castle and the Boyne Valley in Meath. Lovely countryside and Celtic ruins to visit (New Grange) http://www.knowth.com/ .
Have a good trip.