... Maybe not necessarily sights, but pretty areas or locations ...
The 'Levante Haus' - newly renovated. Just have a look inside should you be near it - I think it's great! Location: between C&A and Kaufhof/ Mönckebergstrasse - that is the main shopping street, the Hyatt Hotel is located there, too.
Try and walk around the Alster Lake (the larger part) once - you'll learn a lot about Hamburg! Will take you about 2 hours at an easy pace.
St. Michaelis Church - climb or take the lift to the top - in case of blue clear skies you will have a fantastic view... People here call the church 'Michel' - shortform for Michael.
A Harbour Round trip - even though they removed the "biggest floating picture in the world" (thankx!) - still a must, preferably in one of the small ships... then you get the real feeling of how big container ships can be and, moreover, only the small ships can access the "Speicherstadt", the old part of the harbour with amazing storage space for tea, coffee, carpets...
Every evening in Summer (around August) you can see 'musical fountains' in 'Planten und Blomen' (the gardens at Holstenwall). The lake in those gardens is the scene for light, sound and water and the show happens for at least 20 minutes. They try to make the various fountain movements match the music, they're quite successful. Pretty!
Concerts in the 'Stadtpark' are worth visiting in summer, in case there is something on you like, the open-air setting is great with fantastic acoustics.
Take a walk a little downstream from the Övelgönne Museum Harbour - you'll wonder why the beach suddenly gets crowded (especially in summer) - you'll hit the 'Strandperle' - a simple cafe, more or less a wooden box actually - but highly popular! Taking the walkway instead of running across the beach shows you lots of pretty old little houses. And if you walk far enough, you will reach 'Teufelsbrück', from where you can take the boat back to the centre.
Walk around Eppendorf (where Lehmweg crosses Eppendorfer Landstrasse), Rotherbaum (Milchstrasse, Mittelweg) and the University area (Grindelhof).
The Ice-Rink in Planten and Blomen (behind the Museum of Hamburg History, also worth having a look inside): between November and March you can a) ice-skate or b) watch all the others (elevated good view!) sliding around trying to keep their backsides off the ice. In Summer parts of ice-rink turn into a (roller)blade area. From the ice-rink it's not far to walk down to the harbour or walk all across the Planten and Blomen Park itself.
"Altes Land" - an area south of the river Elbe with (I read that somewhere) the biggest joint area of fruit trees (mainly apples) in Europe. Just don't ask me whether the last sentence is correct English. However, all the villages around the town of "Jork", slightly southwest outside the border of Hamburg, on the way to the city of Stade, are the area of the "Altes Land". From Hamburg you can ride your bike down to the harbour and take the ferry no. 62 to "Finkenwerder" (click the map), taking the bicycle on the ferry is free. When you get off the ferry ride down the dyke (general direction: right) and after a few miles you'll see the first apple trees. Just cycle through the countryside, it's possible without a map. In case you have a map, I was very impressed by the village of "Estebrügge", also "Königreich" and "Jork" are very pretty villages.