Itinerary Suggestions: South and West Ireland

Fly into Dublin and take a direct bus from the airport to Galway city (3h). You can take a bus to Rossaveal (1h) and a ferry to Inis Mór (1h 30m).
Rent a car in Galway and drive to Killarney National Forest (3h). From Killarney drive to Cork City (1h 45m) and make Cork your home base to explore Cobh (45-minutes).
You can fly nonstop between Boston and Shannon Airport that’s closer than Dublin.
When I was in Ireland, I spent the night at a castle in Tarbert. I don’t recommend it though because I was bored since there was nothing to do but have dinner that was OK.

A couple of cool things to see around Cork are:

• Blarney Castle (of course)
• Cork City Gaol (jails are like national monuments in Ireland; so many of their national heroes were rebels against England)
• Murphy's Stout Brewery Tour (unless you are doing the better Guinness one in Dublin)

Around Killarney, there is:

• Muckross House (think Downton Abbey, but in Ireland, in the Killarney National Park)
• Torc Waterfalls (nice nature walk, in the Killarney National Park)
• the Ring of Kerry (a half-day scenic drive around the coast of Kerry, starting and/or ending in Killarney. Don't miss going through the Gap of Dunloe)

Since you like food/drink/music take a look at a couple of nights in Doolin - quaint with several Irish pubs and trad music. Also, the Cliffs of Moher are right there and definitely worth a visit. We took the ferry from Doolin to Inis Oirr for an afternoon but since you're going to be in Galway I'd go to the Arans from there because I understand the ferries are larger with more choices and are less weather dependent.

Will you have a car, or rely on public transportation? If you intend to rent a car upon arrival I strongly suggest you drive no more than an hour after an overseas flight. Driving while jet lagged is dangerous. Do you intend to spend anytime in Dublin?

Would help to know where in the US you'll be coming from.
Also, do you plan to rent a car and, if so, will this be your first time driving on the left?

I've updated the original post, but I'll also reply here to answer some questions. Thanks for everyone's input thus far!

We will be originating out of Indianapolis, Indiana, so we should be able to connect to Boston, New York, over even up through Canada, should that prove beneficial (though with Covid, the fewer hops between countries is likely better). This will be my first time driving on the left, but I'm very familiar with driving a manual. We'll likely spend very little time in Dublin, with any significant time being at the end of the trip (I'm planning on still needing a negative rapid test prior to leaving, so I expect one overnight in Dublin). Should we land in Dublin, we'll likely immediately grab a bus/train to Galway and hire a car there. I've seen suggestions about flying into Shannon which I hadn't considered, so I'll need to give that some thought.

Since you don't seem to be that interested in Dublin it would make more sense to fly in and out of Shannon. It's a lot closer to the things you want to see and do. If you do fly into Dublin and then head to Galway, take the bus, not the train. The bus picks you up outside the terminal. For the train you would have to take a bus into the City Center then catch the train to Galway.

Regarding Killarney National Park. That is easily worth one full day, especially if you enjoy the outdoors.

In numerous trips we have never had a problem getting an automatic car. In my experience, I have driven a manual in the US for years but I could not get used to it in England. I had a very difficult time with all the gears especially reverse and down shifting around traffic circles. I would highly recommend an automatic. Also the idea of Shannon is excellent especially as you seem to be focusing on the west coast. Blarney Castle is a matter of opinion as to is it worth the time but only you can decide if it is worth it. Whatever you do decide you will.enjoy that wonderful country and its delightful.people. fyi We were in Doolin several times and the ferry to the islands were not running due to rough seas, once in October and another time in March. Couple of other times they were. Very weather dependant.

Even if you're comfortable with a manual transmission I think you'll probably be a lot more comfortable opting for an automatic for your first adventure driving on the left. Having to master left-hand shifting while dealing with unfamiliar traffic patterns, narrow rural roads, and generally dealing with unfamiliar rules of the road isn't something I'd recommend for a first timer.
While flying into Shannon would make the logistics a little easier, it looks like it's much easier and cheaper to just fly into Dublin from Indianapolis. From there you can catch the direct bus to Galway as has been mentioned, get a good night's sleep and maybe shed some of your jet lag, then collect your car the next morning. A good base for Connemara is the market town of Clifden located about an hour to the northwest of Galway. From there you'll be central for the very pretty Loop Drive, as well as being close enough to the ferry landing at Rossaveal for a day trip out to the Aran Islands. Flying over is another option if you want to add a scenic flight into the mix.
If the weather cooperates a very pretty drive is the N67 south from Ballyvaughan to Kilkee - lots of terrific photo ops there as well as great views of the sea cliffs all the way down. Plus you wouldn't have to deal with the crowds at Moher.
Killarney town is a congested mess most of the time - especially in June. Getting in and out of there can be a real pain when the tour buses are running. I'd suggest instead basing yourselves somewhere along the stretch between Glenveigh and Castlemaine, from where you could explore Dingle, the northern part of the ROK including the Skellig Ring Road, the Gap of Dunloe and a quick day trip into Killarney to explore the National Park.
There are lots of pretty towns and villages in Ireland but Cork isn't one of them. Unless you have a particular reason for basing yourselves there I'd suggest instead the very pretty little seaside village of Kinsale, from where it's an easy drive over to Cobh. Kinsale can get crowded too, but the workaround is to get an early start to your day to avoid the day trippers who start to show up around 9 AM. Things generally mellow out quite a bit after around 4:30 PM once the last of the tour buses departs.
The Irish Landmark Trust has some unique properties around the island, including a few castles. Might check to see if any of them would be of interest: https://www.irishlandmark.com/.

If you end up flying in and out of Dublin one advantage of renting a car at the airport and driving directly to Galway is the highway is 4-lane and divided - a great opportunity to get comfortable on the opposite side of the road. BTW - you're likely gonna pay more for an automatic. Also, several times after a long flight I've driven without any problems - including to Doolin after landing in Dublin (we flew from Indianapolis and connected in Philadelphia). Sláinte!

As per KBK, I've never had any issues collecting a car upon arrival at Dublin airport and heading out immediately either - I find that I'm usually running on adrenaline for several hours anyway and so am not hesitant to get behind the wheel. Then, too, I've done this several times before so the learning curve of settling back into LH driving isn't a problem for me. Everyone is different, however, so I'd suggest doing an honest assessment of what your level of alertness and your level of comfort are likely to be upon arrival and then plan accordingly.
An automatic will indeed be more expensive than a manual shifter but will frankly be worth it for a first timer IMO.

I've never had any issues collecting a car upon arrival...and heading out immediately

Neither did friends of my sister and brother-in-law who had done this more than once in the past. Shortly after leaving Heathrow he totaled out his rental car, luckily no one was hurt.

While we all agree that most people shouldn't be driving after the long overnight from the US, that advice doesn't necessarily apply to everyone. Some folks are able to get a reasonably good night's sleep on the airplane going over and arrive refreshed, alert, and ready to go. If those lucky few have experience or are otherwise comfortable with LH traffic then the advice to avoid driving upon arrival doesn't apply, which is why I suggest the common sense approach of doing an honest assessment of what an individual's fitness to drive is likely to be and planning accordingly.