Here's what we did:
Flew into Shannon, traveled to Adare Village for coffee and also visited Adare Manor (hotel w/ pretty grounds), then continued to the Rock of Cashel for a tour, then went south to Kinsale (stayed at the Old Presbytery Inn.....LOVED it...request the first floor room with the little balcony looking over the private parking courtyard....very quiet). Then continued toward the coast toward Dingle.
Without digging out a map to remind myself, so the order might be off on the following, we saw the following (and more that I don't recall the names of right now): Dingle, Caha (sp?) Pass (lovely), Connor's Pass, Kenmare, Muckross House (wish we could have spent more time there...beautiful), then on to Dromoland Castle for a two-night stay (don't miss the lovely walled garden, and it's nice to have one of your dinners there.
Then we went over to Doolin to see the Cliffs of Moher (stayed at a small B&B), then the next day headed north after a visit to The Burren, stopped in Galway for lunch (walked along Key Street, full of street performers, cute shops), then headed to the Connemara region (which is an absolutely lovely drive), stopped by Kylemore Abbey (we were short on time, but pace yourself so you can visit it...looked absolutely lovely). If I were to do over, I might add a night in Galway so as to arrive in the Connemara region earlier.
Then after the lovely drive thru the Connemara region, we had a two-night stay at Ashford Castle (which was incredibly wonderful). We made advance reservations for the Hawk Walk, which is an experience where one learns to launch and land their own hawks, after learning a bit about how the hawk lives and eats, etc. That experience is organized thru the School of Falconry at Ashford. We enjoyed most meals at Ashford, but walked into the village of Cong the first night for a meal at a local pub.
For a castle, it's a hard decision between Dromoland and Ashford...both are different. Try to do both (check their web sites for specials, and if they don't have one posted for your time, email them and likely they will offer a special to you), but if I had to choose just one, I would go with Ashford Castle. It's much larger and more castle-like. Both castles offer a variety of activities and both have lakes/water activities, too. We rode bikes one afternoon at Dromoland, which was fun, too.
After the two-night stay at Ashford, we met one of my Irish cousins for the first time, and we were then in their hands for the next two days to see the cottage where my grandmother was raised, etc. in County Roscommon. We overnighted in Tubbercurry at a nice B&B whose owners helped me to locate my cousin.
But, then we returned to the Shannon area the last day to be poised for our flight home. The drive more in-land back to Shannon showed us acres and acres of farmland all bordered by lovely stone fences. You'll see the stone walls all over Ireland, but the drive down to Shannon was different than along the coastal areas. The last night we stayed in Bunratty at the Bunratty Castle Hotel (it is NOT a castle, it is just a more modest hotel, but nice enough, that is located near the castle and close for an easy morning drive to the airport. But, while in there, we visited the Bunratty Folk Park, which as my cousin advised, is as close to what Ireland would have been like during the time in which my grandmother grew up. We thoroughly enjoyed the folk park.
You will have more time that we did, so you can go at a slower pace and absorb more. If I had more time, I would have stayed the a night in the Dingle area, too.
We hired two different drivers (in addition to my cousin's kind hospitality for that of family touring), and if you might be interested in doing that, we can send you info on Tony (who was the wonderful driver we hired for most of the journey).
Enjoy your time in Ireland, it really is beautiful and the people are so very kind.