I've been to Ireland more times than I can count beginning years ago when I had a 4 day layover in Limerick. I've driven the rugged Northern Coast to Antrim and the Giant's Causeway and I've taken the southern route from Waterford to Cork to Killarney. I've stayed in big hotels in the middle of the city, small hotels on the outskirts, B&B's and castles in the middle of nowhere (and that's part of their charm). To me, the secret to really seeing Ireland is to lodge out of the tourist areas and live like a local.
With as little time as a week you can rent a 2 or 3 bedroom townhouse in the lovely village of Adare and explore the most scenic spots in all of Ireland. Adare is filled with thatched roof cottages, beautiful gardens and ivy covered medieval churches. This southwest coast of Ireland is County Kerry and the home to the Ring of Kerry, the Dingle Peninsula and Skellig Michael.
Only an hour and a half north of Adare is Galway, the unofficial arts capital of Ireland. Rent a bike and discover this well laid out city. If you time it right you can go to the Galway Horse Races that are well worth catching. This area is known for its salmon and sea trout. You can fish for it here or enjoy it in one of Galways many great restaurants. And on your way back to Adare stop in Doolin and don't miss the Cliffs of Moher.
When you reside in one place for your entire stay you'll have the opportunity to talk to the locals and find the best place to shop, to eat, to drink. You'll learn your way around and begin to see things you wouldn't have seen if you're just passing through. Play a round of golf on one of the famous courses nearby. Pick up something produced locally from the market for breakfast or lunch. It's a chance to have some memorable down time in a place you've become familiar with while seeing the sights you came to see on short day trips.
You unpack and pack just once and you're not carrying bags everywhere you go. If you're traveling with a family, I can't think of a better option. With a kitchen, living space and at least 2 bedrooms, it's a great place to relax at the end of the day. Saying goodbye to Ireland after immersing yourself in Irish life will be the hardest part of the trip.