Hamburg's public transport is run by the HVV and works well — a single integrated network of U-Bahn (metro), S-Bahn (suburban rail), buses and harbour ferries. The harbour ferries (HADAG) are part of the system, so a normal HVV ticket is enough to ride the line 62 from Landungsbrücken to Finkenwerder, which doubles as a free harbour cruise.
The Hamburg Card is good value if you plan to ride a lot or visit several museums — it covers all transport in the central zone and gives discounts on most museums. Day cards from around €11.
The centre is walkable, and Hamburg has a decent network of bike lanes and the StadtRad public bike-share system, free for the first half hour.