I've been to Copenhagen and no offense to anyone who lives there, but I didn't find the city very interesting...Christiania (a self-goverened free state) within the city was neat, but full of huge dogs, scary looking people and drug dealer types, lol. There's some nice graffiti in there, but that's about it. Everything in Copenhagen was verrrrrrrrrrrrry expensive. The transport links are very good though and you could easily go to Malmö too, which is literally right next door. If you do decide to go to Copenhagen, I would definitely try the smørredbrød (open-faced sandwiches). I had one little one and I have been thinking of it ever since, lol. La Glace is also a nice place to get pastries and the coffee there is really good and it is bottomless (though it is also very pricey...is there a running theme here? lol). There's also a good meatball place in Malmö called Orvars Korvar. And the Turning Torso building is so cool, but it is kinda in the middle of nowhere.
Berlin was a lot more interesting to me but I also found the city quite quiet when I went. There are some interesting museums there though, especially the Jewish Museum. It's also a lot cheaper than Copenhagen and there are more things to see like Reichstag (German parliament), Victory Column (the thing that Bono climbed in U2's "Stay (Faraway, So Close!)" video), Checkpoint Charlie, the Holocaust memorial, remnants of the Berlin Wall, Brandenburg Gate, Gendarmenmarkt, Bebelplatz, Museum Island, etc. When I went there in Feb 2008 there was a free city tour and they brought us all on a pretty comprehensive walking tour of the city. For shopping, definitely check out Kurfürstendamm and KaDeWe. Things to eat in Berlin...umm I think you should try a currywurst. They are not bad and are a local specialty! You won't find them anywhere else. Pork is a big thing there, so if you don't like pork you might starve to death in Germany, lol. For some yummy cheap local food, check out Marheineke Markthalle, which is the cleanest indoor market place I have ever been to.
Can't comment on Amsterdam nor Stockholm as I've never been, but I definitely want to check those cities out sometime! Have fun on your trip!