Twenty or so minutes along the Elbe by car or by S-bahn from the middle of the city lies the lovely riverside suburb of Blankenese. It's nicely situated on the side of a hill, with steep flights of narrow steps that snake between its handsome villas. It makes a popular destination for weekend walks and coffee and cake afterward.
Other parts of town within easy reach of the main station, and worth a visit, include the upscale neighborhood of Eppendorf, its more modest, less self-conscious neighbor Eimsbüttel, and the lakeside suburbs of Harvestehude, Winterhude, and Uhlenhorst.
Farther afield are the BallinStadt emigration museum in Veddel and the Neuengamme Concentration Camp.
Other than driving, the best way to get to Blankenese and Veddel is to take the S-bahn to their respective stations. Neuengamme is reachable by a combination of S-bahn and bus. The other suburbs are no more than 15 minutes away from the center of town by U-bahn.