Well, if you're going to the UK, London is a must: great culture, nice history, good shopping possibilities and so on, however, rather expensive. Manchester could also be nice and I've heard that Edinburgh in Scotland is great as well. But those who live in the UK probably know better as I've only been to London.
About Europe in general... I don't really know about your budget, time or other restrictions and well, it kind of depends on what you want as well so I'm just throwing some suggestions for you.
Vienna, Rome (and other big cities in Italy, like Florence), Budapest and Prague for example are great if you're interested in history and culture.
Paris is a quite traditional traveling destination and brilliant if you're interested in sightseeing and you could visit Euro Disneyland while there. I don't know where you're heading in the UK but it's relatively easy to get to Paris from London by train, though rather expensive. Paris is also the most visited city in Europe traveling wise so it's quite foolproof. So, you can't go wrong with Paris though from my personal experience, the people living there might appear a bit rude as they're quite fed up with tourists. And it doesn't hurt if you know some French as well.
Barcelona, Madrid and Berlin, on the other hand are quite "safe" destinations as they offer something for everyone: night life, culture, history, shopping and so on.
Switzerland (Zurich and Bern for example) is brilliant if you're interested in beautiful landscapes etc. but I've heard that it's a pretty expensive place as well. I haven't visited the place so I can't tell.
And then there are Copenhagen and Stockholm that are quite trendy, stylish cities albeit they might be better places to visit during the summer.
And of course, there's also Brussels, Amsterdam, Dublin... but I don't know about those so much.