Just returned a week ago from our North trip (having done a trip to the South the first time), but we went clockwise, having taken the bus from the Dublin airport directly to Galway to start the trip. Finishing in Dublin worked better for us. But going clockwise, our memorable stops would've included the Stone-Age tomb sights at Bru na Boinne, including Dowth and the Knocknarea Cairns, not just Newgrange and Howth), and we hired local author and guide Anthony Murphy (see his Mythical Ireland Website), who was invaluable with his insight as to their astronomical alignments, links with Irish folklore, and bits that many people would otherwise miss.
In Northern Ireland, we did the unique, reestablished Gobbins Path walk, where a grey seal followed us around the cliff path much of the way,. We also visited the Giant's Causeway, worthwhile despite busloads of people in late September. May shouldn't be too, too crowded, but I hate to imagine July and August crowds. We actually arrived there after a 5-mile hike from the east, which provided lots of stunning views, along with scarily fierce winds. Locals on the hiking path said that day was relatively calm, though!
In northern Donegal, we stayed and ate at The Mill just outside Dunfanaghey. The Dunfanaghey Workhouse, just down the road and described in Rick Steves' book, was quaint but not necessarily worth a 2-hour trip just to see it, but a top restaurant and comfortable B&B made the trip very wirthwhile. Farther South, the Slieve League cliffs and bay were scenic but didn't appear to live up to the "2,000 foot high, highest cliffs in Europe" billing. The low clouds and rain may have blocked some of the views, and maybe we missed a lt of tall cliffs, but they didn't, really make the more-visited, supposedly less-tall Cliffs of Moher, south of Galway, seem that puny.
More Stone-age sights at Ceide Fields, Carrowmore, and Carrowkeel (drive the road up the hill up as far as you can before it's blocked, then start walking), if you're as in to that sort of thing, as are we. And don't miss taking the boat tour of Killary Fjord, and get some mussells on board at the surprisingly good little bar/cafe on the boat, as you glide past the beds where they were raised.
And we did the extended Falconry experience - worth the extra time and cost. After sharing Beckett the hawk between the two of us (everyone can have their own hawk if they prefer, but sharing frees up one person to take photos until you trade off), we also got to fly their cool owl.
Spring for the smallest rental car that will accommodate you and your luggage -- dealing with narrow roads, unforgiving stone walls, and some tight parking spots will reward your choice.