Well if it’s for weekends away I’m assuming Friday, saturday nights and saturday, sunday daytime. Access by rail/air/motorway important. My suggestions are all a bit obvious really (plus massive northern bias):
London’s one but I’m sure that’s already been thought of.
York – history, decent shopping, good pubs, good restaurants, access to the moors/Dales.
Newcastle – Spend a day at the coast or at Durham and then a day in town. Quality food and drink available and the theatre or the Sage have great programs.
Manchester – Great city, loads going on, great bars, great food.
Glasgow – Similar to Manchester really, just a really “vibrant” place.
Birmingham – for the reasons outlined by other folk.
Skipton – Spend time in the dales and nip into Leeds on the train for the day if you want to go shopping.
Ripon – Lovely place for a weekend.