If you'd consider going abroad, Barcelona would be worth considering. It's served by 3 airports (the wonderfully-named El Prat, Reus and Girona) meaning that it's easy to get flights there from anywhere in the UK (and, because of the level of competition, often at really low prices).
Otherwise Norwich is great for short breaks. There's plenty to do and see within the city itself and the Broads and the beautiful North Norfolk coast are all within easy reach.
Canterbury is another possibility, with lots to do in the city and lovely Whitstable just down the road.