Clara, whatever you do don't miss Dingle in Kerry. It's a harbour town and as picturesque as they come. Pop into Dick Macs pub /cobbler. It's famous. The whole peninsula is worth a tour though. There are ruins and the Beehive Huts round about if that's your thing.
And then there's Galway. A don't miss destination for sure. Very bohemian - a university, museum. Good place to stay a few days because there's the ocean cliffs and hills and river valleys all round and plenty to see in the city itself as well.
As an American you would enjoy the beautiful heritage town of Cobh in Cork from where so many American Irish immigrants left Ireland. There's are museums dedicated to the Lusitania and the Titanic.
If you can get to the north go to Antrim and on a fine day take the coastal road from Ballycastle to Larne and visit all the villages in the glens between - Cushenden and Cushendall, Waterfoot. Carnlough is worth a good look round. Glenarriff. All picture book beautiful. It's not far - 80k or so but allow all day because you'll want to stop everywhere. Not far from here is Doagh, the famine village which you might want to visit first depending which way you're going.
Other than those musts.
Kilkenny: wonderful picture book scenery. The town itself has a castle.
I'm also sure you'd like Inis Mor. You can see the whole island by pony and trap which is a unique experience.
Kylemore in Connemara. The Abbey is worth it alone.
I'm sure you've checked out Dublin - Trinity College is a must there.
Those were the highlights for me. You'll love it !