Depends.
If you're female, then it's Edinburgh.Edinburgh has it all. Royal Botanic Gardens are lovely now and all summer. Chambers Street Museum, Museum of Scotland, National Portrait Gallery, Calton Hill (good view from city centre), Arthur's Seat (dormant volcano overlooking the city), Princess Street Gardens, Mansfield Traquair Centre (Edinburgh's Sisteen Chapel!), & the whole city boasts unbelievable stone architecture. And all that's just the free stuff!
Google the images of any of these places!
Edinburgh Castle (amazing view), The Scott Monument (great view),
Camera Obscura (incred low-tech real-time scenes of city), The Zoo, are all reasonably priced. Loadsa shops and cafe's. More pubs per square mile than any other city in Europe, probably the world. Great clubs and nightlife.

Google the images and see for yourself!
During the Festival Fringe...half of London comes up here, tonnes of comedy & music. It's where they all started, everyone from Monty Python to the Mighty Boosh.
If you're male, then Antartica is nice. (Antartica has secret tropical valleys which you'll find eventually if you just keep exploring for a few years.)
In response to: where is the best place in the uk and ire for a week or two break?