You need a bit longer than a weekend for New York City, but it's still a wonderful town. Manhattan is like absolutely nowhere else on the planet I have ever been.
Barcelona is just gorgeous. If you're into good food and drink, this is the place to be.
Rome is the eternal city - a quite magical place. Every other building is a precious antiquity.
Prague is just about my favourite city of all. The first time I went there I spent large amounts of time simply walking around like some sort of demented idiot gazing up at buildings just gobsmacked by the sheer, ethereal beauty of the glorious architecture at every turn. And the beer? Cheap and the the best in the world.
Paris - again, I love the place. Some people think Parisians are nasty folk, but I love them. And again, a beautiful place.
Special mention for Brussels. Some well skanky bits, but the old parts of town around the glorious old square are wonderful, with some of the best restaurants you'll find anywhere in the world. And beer, again - quite superb.