Ireland

Northern Ireland is so beautiful, well worth visiting. Apart from Derry and Belfast, both fascinating, there are lots of Game of Thrones tours of the north coast covering the various filming sites. You could base a tour on their itineraries and add to them, but you’d miss out on the stories that the tour guides tell, which I found really added to the fun of the tour.

Have you seen the Game of Thrones Driving tour maps? Check out here and here. Use them to plan the first part of your trip...

Day 1 Land In Dublin and spend night there to recover from jet lag

Day 2 hire a car (check they are happy for you to take it into Northern Ireland) and head to Belfast for the night, visiting Tollymore Forest Park, Castle Ward and Inch Abbey in the way.

Day 3 From Belfast drive up the coast. Do the slow but very scenic detour round Murlough Head for Murlough Bay and Fair bay. Then head to Carrick a Rede, Ballintoy Harbour (detour for Dark Hedges is a must), Larrybane Quarry and the ruined Dunluce Castle. Plan on an overnight in the Pot Stewart/Portrush/colraine area.

Day4 into Donegal for Slieve League cliffs. Do consider a slight detour to Grianan of Aileach stone fort which is about 8 miles west of Derry. (Map here.) depending on what you are planning to do when you get to Slieve League, this may take most of the day, so plan for an overnight near here.

Day 5 drive to say Killarney - this will take all day and stay there for 2/3 nights.

Day 6 & 7 Ring of Kerry, Killarney National Park

Day 8 Return to Dublin for flight home - OR if it is an early flight, book 2 nights in Killarney and head for Dublin on the afternoon of Day 7 and drop the car off at the airport. Stop the night in an airport hotel.

You will need pounds in northern Ireland and the new visa which you can get on line in literally minutes. Be sure to see the Giants Causeway not too far from the bridge. To the causeway we walked down and took the shuttle bus back up to the visitor center. Well worth it.

I think you're trying to do too much and should choose Northern Ireland or Killarney/Ring of Kerry. If it were me, I would choose NI, spending nights in Bushmills, Derry and Killybegs, divided up based on your interests. If you're worried about driving on the "wrong" side of the road immediately after an overseas flight, an option might be to take the bus to Belfast and stay one night there, picking up the car in the morning. As another poster said, make sure you're able to cross into the border with the car, regardless of where you rent it. Another option would be to get the car on arrival and just drive a short distance north, say to the Boyne Valley. I would not recommend going into Dublin at all on this trip.

side note here when you drive from Ireland into the UK (Northern Ireland ) the road signs change as well as the road markings and the speed limits are in miles in the UK (Northern Ireland) in Ireland they are in Kilometres, (NOTE *when I say Ireland I am referring to the Republic of Ireland whose Official name under that countrys constitution is "Eire/Ireland" ), note advisible to do when you are jet lagged , however if you feel you can then please have a read of the UK's Highway code below these are essential for driving in the United Kingdom
https://www.nidirect.gov.uk/articles/highway-code
do not forget you need a ETA to enter the UK
https://www.gov.uk/eta
do not forget you will need Pounds in Northern Ireland not euro

I agree with others here who suggest just sticking to the north- meaning NI and Donegal in this case. There is so much to see in NI you can easily fill eight days. If you have time, I really recommend a visit to Rathlin Island which isn't that far from Carrick-a-Rede- really beautiful landscape and in August the seabird centre will be open.