Hamburg splits neatly between the formal centre around the Alster and the harbour-side west, and the ferries and S-Bahn make either workable.
Neustadt and Altstadt — the walkable centre
Between the Rathaus, the Binnenalster and the Speicherstadt, so the Elbphilharmonie, Miniatur Wunderland and the shopping streets are all on foot. The most expensive beds, and quiet in the evening once the offices empty.
St. Pauli — the Reeperbahn on your doorstep
The nightlife district, with the Millerntor stadium, the Heiligengeistfeld and the harbour steps all within a few minutes. Beds are cheaper than the centre for a reason: Friday and Saturday nights are loud until dawn, and street noise carries.
Altona and Ottensen — quieter, cheaper, on the water
Residential streets, good markets and a stretch of Elbe promenade, with Altona station on the S-Bahn and the Fischmarkt ferry pier a short walk away. About 10–15 minutes into the centre and noticeably calmer at night.
Practical: stay near an S-Bahn or a ferry pier
Hamburg's water splits the city, and the crossings are the bottleneck. A base within a few minutes of an S-Bahn station or a HADAG pier turns awkward journeys into short ones — the harbour ferries run on ordinary hvv tickets, so they count as transport, not sightseeing.