+1 for the Fishmarket. A lot of people go there straight from a night out. As for a night out you have to try the Reeperbahn, just for the spectacle if nothing else!
In the day time the centre of Hamburg is really nice, around the Alster (the big and little lakes in the middle) is nice. Also take a trip back down to the river front on the U bahn or S bahn.
If the weather is good there are loads of outside beach bars down by the river, sounds wierd but its cool sitting having a beer in the sun with sand under your feet watching the huge cargo ships unloading accross at the docks.