Sounds like a great trip! Some general tips first:
-Don't get a car in Dublin - it will be nothing but trouble in the city. After your time in Dublin go back to the airport - I recommend the bus but with four there may be other options - and rent your car there.
-Pack light for a lot of reasons but also to limit the size of the car you need for four plus bags because Irish roads are narrow.
-If you want to day trip to Belfast you should consider the train and make it adjacent to your Dublin days.
-Read up on the history of "the troubles" to give historical sites - especially in Dublin more significance.
Pay attention to the girls interests (obviously) and limit drive times. About everyone's (on this board anyway) favorite area is Dingle. It is great from a cultural perspective but like a lot of rural Ireland experiencing it requires slowly down and appreciating the quiet. The attraction of this may not go as far with your younger companions.
Some possible sites:
Many people suggest The National Stud which has gardens and horses.
If they have any interest in archeology or ruins the Newgrange/Knowth are amazing and easily reached (<45 minutes) from the airport after the car rental. Also the archeology museum in Dublin is an amazing building with great displays.
The castle in Cahir is large and mostly intact if castle exploring is on the wish list. The smaller castle in Trim is nice (parts were used in shooting Braveheart) and has a great tour and is easily reachable from the Newgrange site.
Dingle, of course, the girls are old enough that they can stay in the pubs to listen to 'trad' music until at least 10 and probably beyond if they with adults and no one is paying too close attention. But there are more adventurous tours too like boat tours to the Blasket Islands from there and you can explore the remnants of people removed from the island - visit the museum before you go. Skellig Island tours are certainly a challenge as well.
The Burren and Cliffs of Moher are the classic West Coast sights and both are worth doing. A guided tour of the Burren might help bring some of the charms of this unique landscape to life.
The Killarney National Park (in the southern area reachable from Dingle) is a great sight with tons of picturesque areas to explore as well as the Muckross House tour (nice if you like that sort of thing) and recreations of historical lifestyles in Ireland through time. The park is free (aside from the tours and things) and is well worth checking out - especially if exploring is on the wish list.
Some suggestions to skip:
The "Dublin Castle" tour. Mostly a tour of official government buildings in which important matters of state happened. Not bad but not the 'castle' most Americans think of when they hear the word.
Galway. I know some people like this town and it seems convenient as a hub but it is probably the least "Irish" town in the Republic and it has no street signs at all making driving there very frustrating.
If you haven't gotten the Rick Steves book for Ireland you really owe it to yourself to get a copy. He has recommendations on places to stay and we've never gone wrong with one of his places. Most have been great and the few that weren't were fine. Also he has kids and gives tips specifically for keeping them entertained. If you want to wait for the 2015 updated version see if you can your hands on an older copy and it will give you a real feel for what places interest you.
Have a great trip!