Bremen, Lübeck, Lüneburg, Schwerin are all easy day trips. All are very walkable from the train station. We did Wismar by car with our dogs, so I can't speak to accessibility re: public transport. Other good day trips include Helgoland, Glückstadt, St. Peter-Ording (probably easier with a car). Time of year makes a difference as to which you might choose. Some are fuller days than others.
The city is friggin' amazing. There is no true center, as someone mentioned, but the Jungfernstieg / Alster-Rathaus-Trostbrücke-Nikolai-Deichstraße-Speicherstadt / Elphi-Landungsbrücken walk will keep you busy and is basically one cool site after another. The Portuguese quarter up to the Michel, the Krameramtstuben, Groß Neumarkt, the composers quarter, the Wallanlagen to St. Pauli and the Reeperbahn, Schulterblatt to Rote Flora is another walk for the more adventurous traveler--gritty in places, but still full of history, architecture, and modern trivia.
The Gängeviertel is probably less worth your time as it's quite small and there isn't a ton to see. A ferry ride, however, is a must. Especially out to Finkenwerder and back.
The Treppenviertel in Blankenese leads you down to the beach and the shipwrecks (low tide only) and is very pretty. The Boberger Sand Dunes offer great nature. South of the river you have places like Jork and Stade that are gorgeous. The Fischmarkt in summer (only on Sundays) is best enjoyed at 5am!