Ive been to Ireland a dozen times and the best tour I have ever taken was by this company http://www.overthetoptours.com/
I took the celtic experience tour and it was just the most amazing thing ever!
I think its the american standard to do the Blarney castle tour. Cliffs of Moher are beautiful and worth it if you can get over on that side of the country.
In Dublin Kilmainham Gaol is a a great site and gives you some good history of dublin. Have a Guinness at the Arlington and see the show one night. Stay away from temple bar unless you want to meet up with the rest of the Americans! You could stay in dublin your whole stay and just take day tours out. I usually use Viator.com for my day tours. Hope it helps!
Thanks so much for the info. We’ve been trying to figure out how to schedule everything—whether we should circumvent the island and stay in a new town every few days or what. Staying in Dublin the whole time while taking day tours to other parts of the island is really starting to sound like the easiest way to do it.