The public transportation in Hamburg is done by the HVV. The important thing to know is that you need to buy a ticket before you go on the platform and you don’t need to clock in the tickets, they are valid after you bought them. Don’t go behind the golden lines in the ground without a ticket! For busses you can buy your ticket at the driver.
A 9-Uhr-Tageskarte - a ticket that is valid the whole day on weekends and in large parts of Hamburg - is the best option most of the time, even if you just take 2 rides. It costs around 6€.
Hamburg has one of the busiest ports in Europe (third after Antwerp and Rotterdam). At the subway station Landungsbrücken harbor tours are starting, where you will get information regarding the harbor while driving around with a small boat. Pro tip: There are also HVV ferries starting there which you can use with your HVV-Tagesticket. With one of these ferries you can also drive to the station Övelgönne - where you can take a walk at the small beach of the Elbe river.
On Sunday morning, starting at 5 am, there is the famous Hamburger Fischmarkt which is also a popular destination for people on their way back home after a party night, e.g. after visiting the bars and clubs of the famous Reeperbahn.
The Speicherstadt is the old warehouse district of Hamburg. Even today the old warehouses are still used for goods like cocoa, coffee or carpets. But some of the buildings are hosting attractions for visitors, e.g. The Hamburg Dungeon and the Miniatur Wunderland.
Planten un Bloomen in the center of Hamburg near the subway station Dammtor is a nice park for having a rest in the sun. In the park there is also the tropical house of the University Hamburg. The entrance is free.
From the top of St. Michael’s Church aka Michel you can see all parts of Hamburg. It costs some money to get up to the tower and currently the lift is not working so you have to take the stairs, but the sight is awesome. An insider tip is to visit the Sankt Petri Church opposite of the town hall. It is not so crowded and it is higher than the Michel.