I agree with seaneendubh, going straight into driving after that long haul - well, it wouldn't be comfortable... Roads in Ireland are tricky at the best of times, and dangerous when you're tired (not trying to discourage you here, just want you to enjoy yourselves!). Have you thought about spending your day in Dublin at the start of the trip? Would give you a chance to acclimatise yourselves, and pick up some maps etc, do some last minute research.
I've never been to the Rock of Cashel or the Dingle Peninsula (despite a couple of fantastic backpacking trips from Cork to Galway along the coast, and numerous daytrips - I'm usually based in Bantry Bay where my grandparents used to live) - somehow I always run out of time - so I can't advise you there. They're top of my list for my next trip though!
However, here's what I can say:
Killarney is well worth a visit - if you're interested in castles and Irish history, Ross Castle is an absolute must, and in a beautiful location! http://www.heritageireland.ie/en/South-West/RossCastle/
The surrounding mountains and oak forests are truly glorious - probably not as big as what you're used to in the states, but staggering nonetheless.
The Ring of Kerry is wonderful, but take your time about it, and try to take a few loops off the "main" road (there's a reason why the buses only go anti-clockwise - the road is too narrow in places for them to be able to pass each other - so tough luck if you want to take public transport in the other direction!! I've hitchhiked a couple of times, frustrating because I couldn't ask people to stop to take pictures).
If you can at all, visit Skellig Michael - it's one of two tiny steep islands off the Iveragh peninsula (boats sail from Ballinskelligs, Portmagee, Valencia Island at about 10am most days until October) that stick up out of the Atlantic like jagged shark fins. Little Skellig is only inhabited by birds, but Great Skellig/Skellig Michael was an outpost for Irish monks for hundreds of years (sacked by the Vikings occasionally). It has also had a lighthouse, now vacated. the only people allowed to stay there now are archeologists in the summer, and only 200 people are allowed on the island every day (though I've never seen that many). You leave the harbours in small boats at 10, and after about 45 minutes sail, have 2-2,5 hours before you're called back to the boats - more than enough to climb the ca. 700 steps to the so-called beehive huts (the stone living quarters of the monks, that still stand after all this time). It's an awe-inspiring place! Also lots of birds, if you're interested. http://www.skelligexperience.com/skellig_michael.html
Skip Kenmare - it's very touristy, and stuffed with wool shops selling frumpy jerseys and hats. Nice in small doses no doubt, but rather overwhelming as it is. Accomodation is expensive even at this time of year (I write from experience!). The mountain pass from there to Glengarriff is spectacular though! But don't try it in bad weather - you won't see anything! But then, that goes for all of Ireland. It's a good time of year to go though.
I can recommend Tralee. Pleasant town with lots going on, and Finnegan's hostel is genuinely charming! (are you hostel or hotel people?)
Fast forward between Tralee and Doolin - the Tarbert ferry, as PhilipD suggests, will cut a few miles off the journey. Doolin is nice (Aille River Hostel is a great place to stay, but book a couple of days ahead, it's popular), as are the villages a little further inland. The Cliffs of Moher are overrated and over-touristy. I agree with Seaneendubh on that count. Some fancy underground viewing platform was being built when last I was there, which meant you couldn't even stand on top of the cliff and enjoy the wind anymore - sounds bizarre.
The Arans are a different story. Overnighting is probably necessary, the boats don't go that often. Inisheer is small and bleak (nice beach in summer - chilly water though!), Inishmore is overrun in summer, but about this time it should be better. It seems more furtile than Inisheer, which basically consists of stones and yet more stones. I've heard that the middle island, Inishmaan is the place to go. Fewer people have discovered it, and it has its fair share of interesting places. Most of the ferries that go from Inisheer to Inishmore do stop at Inishmaan even if it doesn't say so on the schedule (I was cheated by that last time) - ask the skipper. If you're at all interested in literature, try the plays of J.M.Synge, one of the first to properly capture the atmosphere and dialect of western Ireland. He is to the Aran Islands what James Joyce is to Dublin.
If you do go to Inishmore, remember that while everyone else visits the famous Dun Aeonghus on the cliffs, Dun Eochlan inland is also worth a look - there is no official path up there (though there's an official tourist sign when you get there - typical Irish logic;), but with a couple of sturdy trouser (against the brambles) it's an easy walk up from the road - and it's very unlikely anyone else will be there to spoil the view! I went up there one night at full moon, and watched the moonlight shimmering on the sea all around - magical!
Galway is a lovely place to finish off - lots to see, and lots going on, plus good food! There is a night bus to Dublin if you like - takes about 3,5 hours, to spare you the drive. (Also means you can have a few at the pub before leaving, without DUI;)
Phew, long reply - are you still with me?
This would be my suggestion, weather permitting:
Saturday: Arrive Dublin, spend the day and night there. (it's saturday, should be fun!)
Sunday: Drive to Cashel, poke around, continue to Killarney in the evening.
Monday: Explore Killarney, Ross Castle, and surrounds.
Tuesday: Up early, drive south part (so you get the sunset over the water in the evening) of RoK before the roads get busy, take boat to Skellig Michael at 10 (having booked beforehand), afterwards continue to Dingle peninsula.
Wednesday: Dingle, then the long drive north to Doolin. Visit one of the local pubs that have live music.
Thursday: Sail to the Aran Islands. Hire bikes and explore whichever island you've chosen.
Friday: Return to mainland, drive via the Burren to Galway.
Saturday: Explore the various historical places in the vicinity.
Sunday: return to Dublin...
It should be do-able... You'll never have enough time to do all you want, but this should be a good sample taste... I don't have a driving license myself, and envy your mobility! Hitching is easy in Ireland, but restrictive all the same.
Hope you have a great time and that you can use this reply:)