I'm biased as my mom is from Donegal and I've been to Ireland 25+ times, and it's like a second home. So here it goes.
One day in Dublin on the way in or out is enough. Guinness, Temple Bar, Jameson, Teelings, hop on hop off bus.
Skip the south and hit the west and north.
Driving is definitely recommended over bus or train. Like others have said driving can be a challenge on the small back roads (single lane with a bus coming at you). Below is a driving tour that I've done in 4 days. We did it quick and just touched on a few places as I'd been to some before. You cold fly into Shannon and out of Dublin though that may cost more for flight and care rental. If you fly into Dublin and drive across it will take your 3-4 hours to get to Bunratty/Dingle. Take your time.
Bunratty
Dingle
Lahinch
Cliff of Moher
Galway
Westport
Carrick-on-Shannon
Lough Key
Boyle
Sligo
Mullaghmore
Donegal town
Killybegs
Aran Island
Buncrana
Port Rush
Bushmills
Belfast
Dublin
I spend most of my time around Sligo county. Here are some things to do while in the Sligo area:
- Strandhill beach, sand dunes (HUGE), surfing, golf, and The Strand pub,
- Glencar lake and waterfall
- Lough Key / Forest Park
- Carrick-on-Shannon boat tour on the Shannon river
- Hike up Knocknarea or Benbulbin
- Walk through town / pub crawl. Check out McGarrigles and Shoot the Crows. Both are favorites of mine. Connolly's is an old pub, 150+ years I think.
- Mullaghmore head and marina. Great scenery and sometimes offshore big wave surfing.
- Rosses point - Austie's or Harry's pubs. Golf course
My cousin does outdoor yoga, which may not be your thing, but they'll hike to tops of hills and mountains then do their yoga. This is in Sligo town area.
https://www.yourwellbeingwarrior.com/eventscalendar/Also have friends that run Northwest Adventure Tours.
http://northwestadventuretours.ie/ They may have something that interests you.
Slieve League is definitely a place to stop. There are a couple neat towns to stop in. Killybegs is a neat fishing port that some times have the massive "factory" fishing ships in port. Ardara (Are-draw) is another nice sea side town.
Glenveagh park is cool. I haven't been in 20 years but it was always a neat place to go.
The ferry from Rathmullen to Buncrana is something different to break up the driving and see the countryside from the water.
Actually, I'll be there for three weeks in late June and July for my sisters wedding. When are y'all going?