Brian,
Two years ago I spent a week in Ireland with a judge friend who I would consider well off. He has a tour guide (Jimmy O'Leary -- I can find an e-mail address for Jimmy if you're interested) who is the absolute best. Jimmy knows every course operator, innkeeper and restauranteur in the country, both countries. Going with Jimmy is first class. We stayed in great places (Slieve Donard Hotel, Portmarnock Links Hotel, the cool place on top of the hill overlooking Portrush), single rooms, were driven around in Jimmy's huge Mercedes bus, and the food was fabulous everywhere we ate. Jimmy gets great rates and I would highly recommend him for a trip around Ireland and the North.
As Tom Huckaby notes, if you have one hub on the southwest coast (near Shannon), you can play Ballybunion, Lahinch, etc. Then north to Portrush area, with perhaps a stop at County Sligo (near Portrush I'd throw Ballycastle in as a hidden gem along with Castlerock). On way down to Dublin, a stop at County Down, the in Dublin you have Portmarnock, Island, Baltry, even the rabbits and Royal Dublin? Personally I'd skip European Club, but you should give Mr. Ruddy's course a try if you haven't been there.
Then you could, leave Jimmy, ferry across to Scotland with an excursion to Southerness, spend the night at Turnberry (massage and gravity pool, then bagpipes while you're down on the pitch and putt at the bottom of the hill. . . sweet), then Prestwick area (Troon, Western Gailes and Prestwick), then across the country to St. Andrews and a visit to Muirfield area.
Bring plenty of advil, lots of money, and enjoy. . .