Ireland is small enough that a two-week trip can take in both the cities and a serious amount of countryside. Here are the places most worth prioritising.
Dublin deserves at least two full days. Trinity College and the Book of Kells, Kilmainham Gaol, and the National Museum (free entry, excellent Iron Age and Viking collections) are the essential sights. The Guinness Storehouse is more enjoyable than you'd expect from a corporate attraction. In the evenings, the streets around Mulligan's, the Long Hall, and Kehoe's — all on the south side — are better for traditional pubs than the tourist strip of Temple Bar.
Kilkenny is Ireland's most intact medieval town — a Norman castle, a cathedral, an abbey, and a grid of narrow streets still following their 12th-century layout. It is easily visited from Dublin (two hours by bus or train) or as a stop en route to Cork.
Cork rewards a day or two. The English Market is a Victorian covered food market and one of the best in Europe. The city's hills and the River Lee give it a different physical character from Dublin. Blarney Castle is 8km outside the city — yes, people queue to kiss the stone, but the grounds and tower are genuinely worth seeing.
Galway is the western gateway. The city itself is compact and walkable with good pubs and restaurants; the Aran Islands are accessible by ferry from Rossaveal (40 minutes), and Connemara — bog, mountain, and coast — is right on the doorstep.
The Ring of Kerry is the famous loop drive around the Iveragh Peninsula in County Kerry. The scenery is spectacular, and the Skellig Islands visible from the coast are extraordinary. The road is heavily trafficked with tour buses in summer; cycling it (anticlockwise) or driving early in the morning is far preferable.
Glendalough in the Wicklow Mountains is an easy day trip from Dublin that most visitors miss. A 6th-century monastic ruin in a valley with good walking trails — and far more atmospheric than the coach-tour crowd it attracts might suggest.
Newgrange in the Boyne Valley is the prehistoric monument that defines Ireland's deep past. The passage tomb dates to around 3200 BC and the solstice alignment is precise to within minutes. Visits are by guided tour from the visitor centre at Brú na Bóinne; book online.
Donegal in the far northwest is the most Celtic of the counties, with large Irish-speaking communities, some of the most dramatic coastal scenery in Ireland (Slieve League, the highest sea cliffs in Europe), and a compelling emptiness that the more visited southwest lacks.