This may be the most touristy thing you can do BUT you’ll hardly get a better crash-course to Hamburg;
Go to the central station (Hauptbahnhof) and exit direction Kirchenallee. Along the street you will see the busses. You can choose between a few different tours and there is always a buss waiting for passengers. During the tour you can get on and off the buss as you like.
By the way, at the central station you’ll ofcourse also find the tourist office.
The great public transport network stretches throughout town. It takes you almost everywhere!