The Island and Baltray are well worth playing. The Island is quite quirky, starts with 8 straight par 4's, all play differently though, 14 is the narrowest hole you'll ever see, 13 and 16 are two of the best par 3's in Ireland. Baltray has a wonderful variety of holes with great turf.
In Dublin I would look to play Portmarnock, Royal Dublin (which has just finished a renovation), The European Club, just south of Dublin, is well worth a trip. If you are just looking at links golf Laytown and Bettystown between The Island and Baltray should be considered.
In the North RCD and Portrush are exceptional, Portstewart and Castlerock are very good. The parkland courses of Belfast Malone, Belvoir Park and Royal Belfast are very pleasant.
In the North-West the links of Ballyliffin, Rosapenna, Portsalon, Murvagh and Rosses Point, Enniscrone, Carne (which are a bit further west) are all worth looking at.