Depends on what you like, really.
Belfast is a very interesting city with lots to do for pretty much all age ranges, and Derry (or Londonderry to some, still) also has somegood stuff to do. I really like the North Coast (up around the Causeway - the people were so friendly and couldn't do enough for you), and it's so beautiful; Donegal (see same comments as Causeway - truly lovely, not crowded at all, great people and food), and in the other direction, down towards Galway.
Galway's also a lovely town though getting quite pricey the last time I was there and finding reasonably priced accommodation at short notice was really difficult. Great atmosphere and very much a university town these days.
I personally wasn't overly keen on Waterford and Wexford, but then I haven't visited that area since Foot & Mouth back in the early 2000s, when it was made very apparent they all thought we, as English folk, were personally to blame for the outbreak....waking up on our second day with a black mark on the back window of the hire car that hadn't been there the day before was a wee bit unsettling....
Oh, and definitely take layers. And wool. Lots of wool. And waterproofs. Several sets. I have photos from late May a few years ago at the Causeway. I think we all ended up wearing about 6 layers and we were still freezing. (And I'm from Newcastle originally so it's not like I'm not used to cold and wet!!) 