Ireland: Dublin, Aran Islands & Scotland: Edinburgh

You could streamline your logistics a bit and save a day's travel backtracking to Dublin if you opted for an open jaw flight plan (or multi-destination in airline lingo) by flying into Edinburgh and flying home from Dublin. It's a short and cheap flight from Edinburgh to Dublin and from the airport there's an express bus that'll whisk you to Galway in about 3 hours.
I personally am underwhelmed by Moher, but a trip over to the Aran islands from Galway would be time well spent. There are ferries that depart from Rossaveel just north of Galway or seasonally from Galway town itself that include a drive by the cliffs of Moher ... which would be a good way to check that box.
There's also the option of flying over to Inismore from Galway if you don't want to risk a potentially rough crossing on a relatively small vessel ... the boats are about 100' in length and to tend to rock and roll in rough sea states.
You can do all of this without a car by selectively utilizing day tours to points of interest that appeal to you.
Given your limited time I'd suggest forgetting about Belfast this trip.

I agree with the suggestion to fly to Inishmore. We took the first flight of the day over and the last one back, and rented e-bikes to get around the island. It was so much fun and a highlight of our 10 week trip. If you don’t want to ride a bike there are small van tours and horse drawn buggies as an alternative.

My husband and I have about a week...

Edinburgh, Dublin and the Arran Islands. And the Cliffs of Moher. And Belfast.

Step 1: Please define "about a week."  Be specific (it matters) - exactly how many full, usable days do you really have on the ground in Europe? (Don't count a day if you are arriving from, or flying back to, someplace across an ocean on that day, no matter what time the flights arrive or depart.)
Step 2: Recognize that you will burn most of a day when traveling from any of the places you name above to any other.
Step 3: Recognize that you really need to be in or very near your departure city the night before you fly home.
Step 4: Do the math.

I think you will quickly see the issue.

One option would be
Day 1- Arrival Day- to Galway
Day 2- Aran Islands
Day 3- Moher by bus then to Shannon Airport by bus for Ryanair to EDI on TuThSaSu- the Thursday flight usefully leaves as late as 2245. Tram or the 24/7 bus into Edinburgh.
Days 4 and 5 Edinburgh
Day 5 (Saturday)- the 1915 Aer Lingus flight to Belfast City Airport (so you have until about 1600 that day for sightseeing).
Ryanair have an even later flight to Belfast International Airport which arrives at 2205, bus into the city.
Day 6- Belfast then the last train to Dublin. Or bus- the buses run 24/7.
Remainder of the time in Dublin

A lean, time efficient itinerary.

Very much agree to fly into Edinburgh then Ryanair or Easy jet to Shannon or Kerry airports to see western part of Ireland and travel.to Dublin home. I would not try to add Belfast in this trip. Be sure to allow three hours departure time From Dublin as you clear US customs there. Also you will need euros and pounds for the two countries and also the new UK entry pass to enter Scotland. Be sure to book multi city tickets not one way. Good luck.

These are itineraries that worked well for us in the past:
One week in Ireland: Dublin and Belfast, with a day trip from each (Newgrange, Giants Causeway). Public transit and day tours.
One week in Scotland: Edinburgh, Glasgow, day tour to highlands.

Two weeks in Ireland (with car except in Dublin): Dublin, Galway, day trip Aran Islands, Burren, Cliffs of Moher, Dingle, back to Dublin via Cashel.
I would not try to visit both countries in just one week.