Ardmore Co. Waterford Ardmore is a real gem, but a bit out of the way from everything else you'll want to see when landing in Cork. It is relatively undiscovered by tourism. The Cliff House in Ardmore is one of the best restaurants in the country - it's got a Michelin star. It's also fairly pricey - ~€100/head for a meal. There's also the bar separate to the restaurant which also serves good food, and both have the most incredible view. If the weather is favourable, a coffee on the patio the next morning would be nice too! White Horses is a more low-key restaurant in the village which is also well regarded. I know you'll be jet-lagged, but maybe before leaving the next day, consider doing the Cliff Walk around St. Declan's Well. It's particularly scenic, and even doing a small portion of the trail is worthwhile.
Lastly, the Round Tower in Ardmore is likely the only you'll see on your trip. Inside the old church are a few Ogham stones (or as Katharine calls them, bear scratches) - very special!
Kinsale
Beautiful seaside town, a big sailing destination and has a world-famous golf course. Fishy Fishy has fantastic seafood in a mid-range setting, and there's a fish and chip shop on the same street which is also great if you're after something more low-key.
Baltimore
Probably my favorite town in Ireland, one of the places where the well-heeled Irish go on holiday! La Jolie Brise on the front has great seafood, and both Bushes and Jacob's bar are nice lively pubs next door. From here, you can access the islands off West Cork - Sherkin is closer in, and Cape Clear (arguably the nicer of the two) a little further out. The ferry times might not suit your schedule, but a true Irish island experience (different to the Aran Islands, which are relatively more developed)
Schull
Similar vibe to Baltimore. The setting of a podcast that came out this year called West Cork - a true crime podcast which everyone is talking about seems to have become as popular as Serial, S-Town and the other big podcast hits recently. I'd be curious if anybody is talking about it / capitalising off the new-found fame on the ground there..
Bantry
Lovely sea side town - be sure to stop in Mannings Emporium, a fancy food store that does amazing charcuterie plates and sandwiches. I don't know what's good in the town itself, but a lot of seafood lands in this area.
Glengarriff
I don't know much about Glengarriff - scenic! A lot of people do Garnish Island, very close to the mainland - think it's mostly planted gardens, I was there as a child but don't remember it
Between here and Kenmare you can go direct, but there's also the Bere Peninsula, a scenic coastal loop. Castletown-Bearhaven is the town out there, maybe good for lunch?
*Killorglin *
Inch Strand is the must-see around here, really long scenic beach - seem to recall a lot of people riding along the beach?
Dingle A little more touristy than the rest of the towns you'll visit, but still lovely. This sets you up for one of the main scenic drives out around here. 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, out around Dunquin and back into Dingle via Ballyferriter. The Blasket Island visitor centre is well worth a visit (as are the islands themselves, but it's at least half a day of a trip if not more)
Cork You can head back to Cork via Killarney (and it’s stunning national park - worth some detours). Muckross House (old country house) in Killarney may also be worth a visit. Cork is Ireland’s second city - Katharine preferred it to Dublin. . Corkonians are very proud of their city, referring to it as “The Real Capital” or the “People’s Republic of Cork”. The highlight for me is the “English market” - an old victorian market with lots of interesting food vendors. The Farmhouse Cafe in there is known for traditional food to the market like Tripe (stomach) and Drisheen (a special type of black blood pudding). I did say traditional - which depending on definition may not strictly be synonymous with good :-) A little out of the way depending on where you're staying, but I always liked the Franciscan Well, a craft brewery.