7 Nights - Dublin & ?

Less the 3 nights in Dublin, you have 4 nights total. Depending on the ferry schedule you may not have enough time for the Aran Islands. If you can’t go there, then substitute it with the Dingle peninsula. The National Park, as well as the Cliffs of Moher and the nearby Burren, plus either the Aran Islands or the Dingle peninsula will take up the time you have. The wild card is the walking you intend on doing as it could occupy a good deal of your time. Galway really isn’t that large, at least the city center area isn’t. When we were in Galway, we stayed at one of the B&Bs on College Rd up by the greyhound stadium. Not downtown, but easy to get downtown. On your return to Dublin, if you had time, you could stop at Glendalough for some walking and to visit the Monastic City. There’s a nice waterfall there. If you’re a fan of old UK TV series, Ballykissangel was filmed in nearby Avoca, where there’s a woolen mill worth stopping to see.

Is 4 nights enough for Dingle considering the driving time there and back from Dublin? Just seemed like too much time driving and not enough time to see much.

Dublin to Galway is mostly interstate and should take a couple hours at most. I assumed you’d be driving to Dingle from Galway which is 94 miles. When we were there we left Galway and saw the Cliffs of Moher and drove the Dingle Peninsula on the same day. After driving the peninsula, we spent the night in Tralee. Again, if you go to the Aran Islands you probably won’t have time for Dingle.

One night Dublin on arrival, next morning leave. Head to Glendalough, thru the Sally Gap in route to the Rock of Cashel then onto Cobh or Kinsale for an overnight. Meander about the next day. Next morning off to Dingle via either Kenmare or Killarney.
2 nights in Dingle. Then drive north towards Galway. Maybe an over night in Doolin. Off to visit Aran Island the next day. Return to Doolin. Off to Galway. Overnight then head back to Dublin.

You could ease the logistics somewhat by flying out of Pisa rather than Florence - am seeing a nonstop on Aer Lingus that would get you into Dublin around noon.
Since you'll have already dealt with your jetlag you could rent a car upon arrival at the airport to begin your driving journey or, alternatively, take the train to Galway and collect your car there. The picturesque village of Clifden is located about an hour west of Galway, and from there you'd be central for exploring Connemara, to include the very pretty loop drive via the National Park (nice hikes for all skill levels), Kylemore Abbey and the village of Cong. There's a Hawk Walk at Ashford Castle there that many here rave about, and both Ashford and Cong are pleasant places to wander about for a day. From Clifden you'd also be close to the ferry landing at Rossaveal for a day trip out to Inishmore (or better yet an overnight). A hidden gem that we particularly enjoyed is the short ferry ride from Cleggan out to the island of Inishbofin. Connemara is our favorite part of Ireland - there's plenty there to keep you occupied for several days, and it features some of the most spectacular scenery in Ireland.
The temptation on a first visit to Ireland is to try to pack too much in to a short stay and thus run yourself ragged trying to "see everything". Personally I'd try to guard against that and just concentrate on a couple of destinations ... in this case Connemara and Dublin. There's plenty to see and do in Dublin as well - including day trips and tours to Glendalough and the Wicklow Mtns., or maybe up to Newgrange and the Boyne Valley if you're keen on Irish history.
There are also easy day trips by train up to Belfast or down to Kilkenny.

In May 2018, we landed in Dublin and drove directly from the airport to Doolin via The Burren and spent 3 nights. The Cliffs of Moher are right there and you can access the Aran Island of Inis Oírr via the ferry (weather permitting). Doolin is a cute little town but 4 nights might be one too many. After Doolin we drove back to Dublin via the Rock of Cashel, Waterford, Kilkenny before dropping the car at the airport and then spending our last 3 nights in Dublin. If we would have had a 7th night we would have stayed in Kilkenny before going back to Dublin. My advice would be to head out after landing and don't waste the time it will take to stay in Dublin twice.

Thanks - I think that Galway/Doolin area makes more sense to us than the longer drive to Dingle. If it were just hubby and I, we'd go for it.. however, our friends aren't well travelled and she occasionally has a migraine after plane rides, which is why we're most likely going to stay in Dublin after we land, but I will talk with them to see what they think.

The beauty of Glendalough will ease her migraine. Too bad she’s not adventurous as the R115 ( the old military road) thru the Wicklow Mountains is unencumbered and Gorgeous. HOWEVER, I’d not recommend it this trip. Eats up a lot of time.

P.S. you’ll see plenty of sheep.

As far as small towns stay in Port MAgee instead of Dingle. Kinsale or Cobh.

Choose to not see Dublin when you arrive. Instead spend your first night in Tralee. Easy peasy drive from there to Galway the next day.

Consider Northern Ireland. If you like small towns and you like cliffs and walking, I'd suggest the Giant's Causeway area.
We stayed at the Bayview Hotel in Port Ballintrae (highly recommend), and had great walks to Causeway visitor center, Dunluce Castle and Port Rush.

We spent 3 relaxing and enjoyable nights there.