I'm based in Cork City, and I do most of my riding around Munster, the South West part of Ireland. The most interesting parts of Ireland are probably Dublin City for historical and cultural reasons and the south and west coasts for scenery and tourist atractions. The inland (midlands) parts are nice enough too but could be a bit dull compared to the west.
West Cork, south and west Kerry, west Clare, county Galway and Galway city, Mayo and Sligo are well worth visiting. Donegal is lovely too but is a good distance from Shannon or Dublin Airports, where you'd be arriving into Ireland from the States. It is also one of the worst areas for road accidents, particularly the northernmost area, the Inishowen peninsula. I cycled the length of Ireland 2 years ago for charity and despite the great scenery Donegal was by far the least enjoyable area to cycle through.
I reckon a nice tour would be:
Arrive Shannon
Head north to County Clare: Ennis, Doolin, Ballyvaughan, Then into County Galway: Galway city, Connemara, then Westport in County Mayo
Back to Clare, into County Kerry: Tralee, Dingle, Killarney, Kenmare, then into County Cork: Bantry, Beara Peninsula, Skibbereen, then work back to Shannon across north county cork. I'll take a look at a few maps and see if I can get mileages.
Bear in mind the weather here is never very warm or cold but it does change a lot, we get a good deal of misty rain. Bikes shops are plentiful and well-stocked, accomodation is also plentiful and there are a range of prices, from basic hostels to Bed and Breakfasts and hotels. There are often very good value mid-week specials. There are loads of options for food, most towns have well stocked delis where you can get subs and sandwiches made to order. Restaurants can be pricey but are generally good quality.
Public transport is expensive, dirty and generally crappy, trains won't take bikes and buses will take them at the drivers discretion which is a bit hit and miss.
From a safety point-of-view, the main roads have hard shoulders but these can be dangerous as they are sometimes used as a slow lane by trucks, tractors etc. They also have a tendency to disappear without warning. Most roads have good surfaces but do not have well-defined edges, or markings to denote the edge of the roadway, so the strip that cyclists use is generally in poor repair.
There is a serious problem with speeding and unregulated young/learner drivers at the moment and this getting worse, so be prepared for agressive driving particularly in the outskirts of towns and cities and at night and weekends. I felt a lot safer pedalling north along Route 1 into Boston than on any Irish road.
Let me know when you firm up on plans, dates, routes etc and I'll be glad to give you more details.