Top places/things to see:

Alter Elbtunnel | Landungsbrücken 20359.
When leaving the subway station, you'll probably have the most impressive view of Hamburg with the harbour in the background and the old building - a shipping pier and the entrance of the old Elbtunnel.

Rote Flora | Schulterblatt 71, 20357.
You'll find the Rote Flora - a famous old building that was a cultural and political meeting point in the Schanzenviertel quarter. The Schazen district has lots of pubs, bars, shops, music clubs, and design and artists studios. It is seen as the sub-cultural centre of Hamburg.

Glockengießerwal / Steintorwall - You'll find all important art museums and the main station of Hamburg in this area. The Kunsthalle and the Deichtorhallen for example are very very good ones.

Museums to see:
Hamburger Kunsthalle - Glockengießerwall 1, 2009 | www.hamburger-kunsthalle.de
Deichtorhallen - Deichtorstraße 1, 20095 | www.deichtorhallen.de

Speicherstadt / HafenCity - Where old meets new. The Speicherstadt is the very old and nice warehouse district at the River Elbe. HafenCity on the other hand is Hamburg's newest quarter - where you'll see lots of modern architecture.