Itinerary Suggestions

I would say if you are coming from the USA you should not drive all the way to Dingle after arriving on an overnight flight. The drive to Dingle is 5 hours without stops for food/bathroom/sightseeing. You will be jetlagged. You will be tired. Its really not safe.

Fly into Shannon airport, it’s a very short drive to Ennis, a charming small town. I would spend your first night in Ennis. Take a taxi from Shannon airport to Ennis, about a 30 minute drive.

Then the next day rent a car and drive to Dingle, spend 3 nights there. Dingle is delightful.
Drive the Slea Head drive in a reverse direction from the usual. See the Blasket Island Center, Cliffs of Moher and Gallarus Oratory.
Others will have lots of ideas about the days on your way to Dublin.
Then fly home out of Dublin.

Sounds like a wonderful trip!
Use multi city on your airline’s website to book your flights.

After Dingle, work your way up the western coast, and spend a couple days in Galway. You could/should also make a trip to the Aran Islands - at least see the big island, Inishmore. You can get there by ferry, but on our last trip, we found that flying was the best, fastest, most reliable way. Airport for the small-plane flights is west of Galway.

Agree don’t fly into Dublin and spend 4 to 5 hours driving to Dingle.

Fly OPENJAW: Into Shannon out of Dublin.

Shannon to Dingle is a 3 hour drive.

As far as where else to go, I’d say when in Dingle explore the Gap of Dunloe and Killarney National Park, visit the Portmagee cliffs, and take the Slea Head drive ( go early to avoid the tourist busses) and hike above Coumeenoole Beach to Dunmore Head.

If it’s a sunny day you will see the Great Blasket Island.

Gorgeous country, wonderful people, good food. Drive carefully to avoid dog walkers, hikers, sheep and bicyclists. Slainte

Rather than getting behind the wheel immediately upon arrival, an alternative might be to take the train over to Galway and then get a good night's sleep in a real bed before collecting your rental car there to begin your explorations. Takes about 2 hours.
The Connemara area is beautiful - our particular favorite in fact. Rather than basing yourself in Galway (after your 1st night) you might consider the pretty market town of Clifden about an hour to the Northwest - it's central to the Connemara Loop drive which includes the National Park, Kylemore Abbey and the village of Cong ... where you'll also find Ashford Castle. It's a pricey place to stay but you can book their "Hawk Walk" which many here rave about without actually staying on the property. From Clifden it's a short drive over to the village of Cleggan, from where you can take a short ferry ride over to the island of Inishbofin ... one of Ireland's better kept secrets in our opinion. It's also not far from Rossaveal which is where the ferry over to the Aran Islands departs from.
Dingle is great, but you can pretty much satisfy your curiosity about the place in a day, and it's not a particularly central base for exploring anywhere else. An alternative that we've settled on for our own trips to the area is to find a place somewhere between Glenbeigh and Castlemaine, from where you'd have a short drive up to Dingle, as well as heading the other way to explore the northern part of the Ring of Kerry, Killarney and the National Park, and the very pretty Skelling Ring Road that includes the Kerry Cliffs that someone else mentioned. Valencia Island is quite pretty too- with picture postcard views of the Skelligs.
Flying in and out of Dublin will probably be much cheaper than trying to do some sort of open jaw by flying into or out of Shannon, but it's certainly worth doing a sample booking to see if it would be more cost-effective to do so.

I like the idea of flying into one airport and out the other i.e., Shannon and Dublin. Don’t drive on your day of arrival is crucial advice as others mentioned.
Consider spending one night on Inishmore to appreciate it more; I wouldn’t spend much time in Galway though. From there drive to Trim then on to Boyne Valley before going to Dublin.

Some of our earlier trips we were able to fly into one airport and out of the other but in recent times it wasn't as easy as I seem to remember there were few direct flights to and or from Shannon to the US. The price difference was very minimal. It is certainly worth your checking into this option. if this doesn't work why not start your trip with a few days in Dublin and then renting a car once you adjusted to the time difference and head out to the west coast, make a circle and return to Dublin for your last night. Highly suggest kilmainhaim Gaol in Dublin but these days you need an advance ticket so don't miss out on that. Dingle is worth two days at most. We love it there but there are so many other places to see. Best of luck and enjoy. We stayed at the Lighthouse B and B several times and loved it. It is not directly in town though but we liked the loication. The day you fly out be sure to have three hours at the airport as you clear US customs there and if direct flight arrive in US as if you are on a domestic flight.

I understand a Dublin round trip.

There are plenty of nice little places to stay in Ireland, probably a good number you have never heard of but will drive through - and you can see the most unexpected historical sites just by happening upon them. But if I were picking from the advice you’ve gotten, I would second Robert’s. I have been south twice and not yet to Clifden but it looks gorgeous and it’s on my radar for next time.

I am planning a June 2022 trip and I was able to book my flights with Aer Lingus from BOS-SNN and then from DUB-BOS. I think those Shannon flights are supposed to start up again in April, if I rememeber correctly. Fingers Crossed!

You can easily access the Cliffs of Moher, the Aran Islands (weather permitting), and The Burren from the fun little town of Doolin. Ennis is larger and a little farther away but another good option if you're visiting that general area.

The Irish government used to require Aer Lingus to do a certain number of flights into Shannon. The idea was to sustain and develop that airport. Such is no longer the case and thus there are a lot less Transatlantic flights going through Shannon. It wasn't uncommon for a plane to very briefly touch down in Shannon, unload a few passengers and then wheels up for Dublin with the rest of the Planeload.

I have taken this Game of Thrones tour from Belfast This is a must-see tour for any Game of Thrones Fan. This tour takes you to several of the Game of Thrones filming locations used in the series and also includes a visit to the Giants Causeway the only UNESCO World Heritage Site in Northern Ireland, Even if you’re not into the show, the locations are beautiful places to see. The Dark Hedges One of the most popular natural phenomena in Northern Ireland, the Dark Hedges were used as a set for the hit TV series Game of Thrones. It was the famous King’s Road in Season 2. you can book this tour on https://www.belfasttoursni.com/game-of-thrones-tour-from-belfast/