We've just got back from Prague and Berlin last week.
Prague is best for wandering around all it old town streets, its good to turn off and explore little roads, obviously
explore the castle... to see Charles bridge its best to get up early in the morning, though be warned that they are currently doing restoration work on it.
We accidently found this medieval themed bar - http://www.svabove.cz/ its good place for a beer or 3.
Prague is quite cheap (even though prices have risen in recent years) so you can stay in a decent hotel for not much.
In Berlin we enjoyed wandering around Friedrichshain and Prenzlauer Berg, there's lots of good music, but that depends on what you like- for something a bit different try Kaffee Burger and as you are there on a wednesday check out the knochenbox in the crypt of the Theater/Chape in Friedrichshain.
In Berlin people don't go out until late and places generally don't close until the morning.
In Berlin we stayed in the 'all in' hostel in Friedrichshain, they also have private on suite rooms.
In Amsterdam just wanding around the streets is nice, the sex museum is a must, the van gogh museum is disapointing.
Do Anne Frank's house first thing in the morning and have something fun planned for straight after.
If you like science stuff and acting like a kid, then head to the nemo centre- its lots of fun.
I would advise having tip on a boat- even if its just a river taxi, its a nice way to see the city.
we always stay down by the leidseplein as it has lots going on around it and as it is a walk or very short tram trip from the station its a bit cheaper.
Everywhere will be cold and everywhere will be rainy, you will need an umbrella.