If you just had three days for Ireland.....

Fly into Shannon and enjoy the West Coast.

I’d arrive, get the rental car and spend time driving to Dingle and base there. About 2 1/2 hours without stopping but trust me you’ll stop so adjust your schedule accordingly.

Visit Portamghee to see the cliffs, drive Slea Head, see Killarney National Park, Drive or hike the Gap of Dunloe, visit Sneem.

Have a pint at Foxy Johns in Dingle.

Fly Shannon to Heathrow.

Where ever you land, stay close to there. No real time to explore. So your two likely options are Dublin or Shannon, worse if it is your first stop (jet lag), so plan things close to there...no sense spending half your time driving. Same thing for Scotland, either Edinburgh or Glasgow, probably Edinburgh over Glasgow (for a newby) same with Ireland, maybe Dublin over Shannon, though a toss up.

I much prefer anywhere in the country over Dublin.

Doolin, Co. Clare (1 night); Inishmore (Aran Islands, 1 night) and Connemara (2 nights) both in Co. Galway.

I would definitely recommend against getting a car after a transatlantic flight, especially since driving is on the left.

There's enough in Dublin for 3 full days. Some of the best parts are the less accessible ones and the weather is always iffy. You could visit the Cliffs of Moher as a day trip from Dublin - if the weather's good. When it's sunny, it's beautiful. Probably not so much on a cloudy day, miserable on a rainy one.

I guess it depends on where you enter the country and whether you wanted to see the Republic of Ireland or Northern Ireland. To me Dublin is good for a day. Glendalough and the Monastic City are nice if you wanted to spend time outdoors. If you like castles, there are plenty to visit, e.g., Trim, Cahir. The Burrin, Cliffs of Mohir, Dngle Peninsula, Muckross House, Hill of Tara, and Rock of Cashel are popular places. In the north are the Giant’s causeway, Carrick-a-Rede rope bridge, Carrickfergus castle, the ruins of Dunluce castle, and Bushmills distillery. If you’ve seen the TV show Ballykissangel, it was filmed primarily in Avoca, just south of Dublin. The town doesn’t take long to see. Just outside of town is the woolen mill that is worth stopping at. Of course, there are always the larger cities if you wanted to spend your time in them. Remember that you’ll need British pounds in Northern Ireland and Euros in the Republic of Ireland.

If this is to be your first visit to Ireland then I suggest keeping it simple - ie flying into Dublin and using that as your base for the limited time you have available. There's plenty to see in the city and there are easy day trips available to explore further afield: north to Newgrange, Tara, and the Boyne Valley and south to Glendalough, the Wicklow Mtns. and perhaps Powerscourt. You could also do self-organized day trips down to Kilkenny or up to Belfast for a day - each an easy day trip by train from the city center. Organizing your trip that way wouldn't require renting a car, thus simplifying things for you immensely.
With so much to see and do within an hour or so of Dublin I'd advise not wasting most of a full day trying to make it to attractions on the west coast. Better (IMHO) to save Kerry, Dingle and Connemara for a future visit when you'll have a chance to enjoy the charms of the West of Ireland at your leisure.

I agree with Robert above — if your 3-4 days include arrival in the country and departure day onward, you really don’t have time for these west coast escapades, especially when you take into account that driving upon arriving on a transatlantic flight is dangerous for both you and any other drivers and passengers on the road.

Dublin and outskirts has lot to offer; stick there on such a short stopover.

How about you stop there last just before returning to the states. That way you won't be jet lagged so driving would be safe. I agree driving on the 'wrong' side of the road after a transatlantic flight is dangerous to you and others. Don't do it. However, the west coast is the 'best' part. We had nine days and started in Dublin and went south first, then up the west coast. Those first few days in Dublin and south I kept wondering why people thought Ireland was so great. Then we got to the west coast. Much better. I'd spend all three days there.

I agree with Kim and Robert. I would stick to Dublin and areas if you have 3 days.

Dublin is a good three night stay. Haven't been to Shannon but agree that is the other option with a day trip or two in that area.

First, if you've been to Europe "many times" shuffle your plan to give Ireland (or Scotland) the time they deserve. Ten days in either would give you a good taste but would still be rushed.

Second, I suggest you post your "preferences" and see what happens. What I mean by preferences is that you re-order the following list according to your taste. I've provided example(s) of each just for clarity, not as a recommendation, usually one that I'd consider popular and one not so visited. I'm sure the list is incomplete.

Natural beauty (Kerry Cliffs, Connemara)
Hiking (The Burren, Anascaul Lake )
Music (Cobblestone, The Crane)
Neolithic sites (New Grange, Drombeg)
Castle Ruins (Cahir, Carey)
Restored castles (Dungaire Castle, Bunratty)
Elaborate functional castles (Dublin Castle)
Christian ruins (Clonacnoise, St. Patrick's Well)
Gardens (Dublin Nat'l Botanic Gardens)
Wildlife (Skellig Michael in June, Shannon River)
Religious centers (St. Patricks Cathedral )
Cultural (EPIC)
Cemeteries (Glasnevan)
Museums (Kilmainham, Derryglad Folk Centre)
Other

Hotel versus B&B versus Castle lodging
Self Driving versus Tour Bus/van
Popular versus Off-the-beaten-path (Blarney Castle, Carrowkeel)

We're also interested in the area around Shannon (3 days at end of Scotland trip), but via public transit (bus OK) and/or group day tours. We don't want to rent a car. I saw that there are Gap of Dunloe tours, Additional options? Or are logistics too cumbersome to be worth it without a car?

I would fly into shannon and skip Dublin. Thw Connemara area is beautiful.

For Elaine: exploring the west of Ireland using public transport from Shannon/Limerick would be difficult. I suppose there are local bus routes you could cobble together to see some of the countryside but you’ll waste a lot of time just trying make that work.
Limerick would work as a base if you want to keep the logistics as easy as possible and rely on local day trip providers, which might be your best bet if a car is out of the question.