I’ve tried to split this up into day-sized chunks. Most of Ireland’s scenery can be seen by car for free - unlike the US, beaches and coastal parking lots are always free, and never require residents permits.
Driving Dublin to Galway, there is nothing that I can recall to recommend, ~2hrs good motorway drive.
From Galway, you can join what is called the Wild Atlantic Way - a coastal route following a very well sign posted (and marketed) route both south and north which traverses the entire western coast. Roads are narrow, but cars are small - just takes a little getting used to!
Galway Stay a night or so. Nice old city with essentially one street. Most people’s favourite Irish city. Connemara National Park scenery is stunning- driving the N59 for a day is well worthwhile, turning around in Leenane - wouldn’t go too much further north in a day, made that mistake before! Some stuff to do on this loop (roughly north to south): Aasleigh Falls - actually just over the border into Mayo Killary Fjord - Ireland’s only fjord Glassillaun Beach is a hidden gem, one of my favourites Sky Road in Cliftden - stunning scenic drive. Kylemore Abbey - beautiful old house.
———New day——
From Galway, head south until Kilcolgan. From here, follow Wild Atlantic Way signs to The Burren, entering Co. Clare. Ailwee Caves are very impressive limestone caves. If you visit these, you might need to backtrack a bit to Ballyvaughen, then continue to follow the coastal route out to Black Head. Consider Doolin or Lisdoonvarna for lunch. Doolin has a resident Dolphin called Dusty much of the year. Lisdoonvarna has a wold famous matchmaking festival every year. The Cliffs of Moher is Ireland’s most visited tourist destination, and Europe’s tallest sea-cliffs. €6/person to enter - the one much-contested exception to the rule of “everything is free”., but worth visiting.
From here, I’d leave the coast & take the quickest route South, via Limerick City. On the way into Limerick is another very popular tourist destination, and probably the best castle to visit in Ireland: Bunratty Castle & Folk Park. The folk park is very tastefully done & authentic - plenty of Irish visit here every year too! Limerick is (by Irish standards) a “rougher” city, but this is relatively speaking - still quite safe, but I usually skip it on any tour. From Limerick, head for Dingle - charming West Kerry town with lots of pubs and restaurants. Dingle is worth staying in.
——New Day—— From Dingle, follow Slea Head Drive entire way around the peninsula. This drive has the most spectacular coastal scenery I’ve ever seen, I think the best in Ireland. This drive is a big loop, ending back in Dingle. From here, head via Killarney (and it’s stunning national park - worth some detours) towards Cork. Muckross House (old country house) in Killarney may also be worth a visit. Cork is Ireland’s second city. Corkonians are very proud of their city, referring to it as “The Real Capital” or the “People’s Republic of Cork”. Worth staying overnight and wandering around, the highlight is the “English market” - an old victorian market with lots of interesting food vendors.
From Cork, I’d skip the South East (where I’m from, so I’m allowed to say that!) and head straight back to Dublin.