It depends what you are looking for? Budapest has some lovely parts, and not so lovely, there's a lot of poor modern building in the centre of the city that detracts from the vista, but equally there are some very attractive parts and there has been a lot of renovation in recent years. I was last there three years ago (stayed at the Kempinski) and did find dining options limited, the shopping not so good, but did enjoy it overall - especially in the old town around the cathedral (the Hilton there is well sited).
Prague on the other hand is just so beautiful in winter. The lanes and streets are enchanting, there's so much to see and do, great restaurants, shopping, arts and a visit to the Jewish quarter is fascinating (and torrid because of the history), watching the changing of the guard at Prague Castle is like something out of ruritania!
Copenhagen has different qualities, a beautiful setting, another fantastic city, but like anywhere in Scandinavia you might have wide eyes at the price of things ...
Not to confuse you more but three other places in Europe I find wonderful are Stockholm (eye wateringly expensive though), Kracow (probably my favourite place of all) and Dresden which I also thoroughly recommend (ensure you book tickets for the Green Vault in advance).
The no. 1 and I take a trip somewhere in easter Europe every winter, this year was Venice which doesn't really count, and in February it'll be Munich I think. Many of the cities are wonderful and if you don't mind the cold winter is a great time to visit, out of season they're quieter and more enjoyable (Prague in particular can be heaving in summer).
There's a plane at JFK, to fly you back from far away
all those dark and frantic transatlantic miles