Hamburg's entertainment district is called the Reeperbahn. The last time I was there, Beatles clubs like the Indra Club, the Kaiserkeller, the Star Club and Top Ten were still operating though sometimes not exactly in the same buildings they occupied in the '60s and I think one of the clubs is now called 64. There's also a Beatles-Platz at the beginning of Grosse Freiheit where the clubs are located. There is more on Grosse Freiheit besides the Beatles. Much more. (Tip: Pay cash when you can and hold on to your credit cards).
Sightseeing is great. The City Hall square has vendors and performers, there is shopping in the city center. My favorite was a short cruise on the Elbe River.