I’m not from the auld sod but I lived there for a long time.
I’m not a fan of Cork city. It’s OK for half a day, the English Market’s nice, have a pint of Murphy’s and then… meh.
If you’re renting a car, I strongly recommend driving a few hours further to West Cork - Castletown Bere area - or even driving north and around the Ring of Kerry or the Dingle Peninsula. Dingle’s my favourite place in Ireland.
Definitely recommend Newgrange. It’s amazing. Make sure you go early in the day so you can do the tour.
Jonny Fox’s in the Dublin Mountains is a huge tourist trap, but it’s in a great setting and does fantastic seafood.
Dublin:
Mulligan’s of Poolbeg Street is a historic pub with reputedly the best Guinness in Dublin.
Go and see the Book of Kells in Trinity.
The Literary Pub Crawl is a riot. One of the most fun evenings you can have as a tourist.
The Dawson Lounge on Dawson Street is worth a visit. It’s absolutely tiny.
The Cobblestone in Smithfield has ‘real’ live music. People playing spontaneously because they want to, rather than because they’re being paid to go diddly-idle-dee for the tourists. It’s where Billy Connolly played when he visited.
I’ve been out of the country for so long I’ll leave the rest of the recommendations to actual Paddies, as I’m no doubt out of date.
Notes about driving:
If you’re American you’ll probably want an automatic. Make sure you book this well in advance as they’re quite rare.
Practice driving on the left in the side roads around the car rental place for about ten minutes before driving on the highway. It’s quite easy to adapt, but make sure you have a significant aide-memoir in the car that reminds you to set off on the left. If you’re distracted it’s easy to set off on the wrong side. My dad has totalled a car doing this in the US, I nearly did too when I was living in the states, and friends of mine from Texas lost a week of their vacation in hospital in Galway with broken bones following this error.
Though the distances between towns and cities seem small, the driving conditions are poor. Dublin-Galway and Dublin-Cork now have toll motorways (freeways) between them most of the way, but elsewhere you’re looking at an average speed of about 40mph long distance.
Finally:
Bring an umbrella.