Hamburg is a great place for outdoor activities. The Lake Alster is the hot spot for runners and if the weather plays along for long walks, biking, canoeing, kayaking, and sailing.
Same with river Elbe that offers a long promenade to walk, bike or run.
Golfers, in particular from North America, might be disappointed with the golf courses. Not because of their quality but because of the requested golf license. Most clubs don’t allow any guests that do not belong to a golf club elsewhere in the world and have a club managed handicap. Sometimes you are even not allowed to play although you fulfill the restrictions because your handicap is too high. I only recommend playing in Germany if you cannot live without a round of golf for several days or weeks to save yourself a ridiculous hassle.
If you are more into watching sports you should think of getting tickets to a soccer game. In Hamburg there are two clubs whose fans are incredibly antagonized with each other. One is the Hamburger Sportverein (HSV) playing in the 1st Bundesliga and the second is St. Pauli playing in the 2nd Bundesliga (according to the season 2011/2012). I might get myself into trouble describing those two clubs but St. Pauli attracts more or less the left wing with a lot of pot smoking followers while HSV fans are more center-right minded.
HIKING, BIKING, RUNNING
LAKE ALSTER If you want to do some walking, biking or running around Lake Alster (a total of almost 5 miles) you could start from metro station Klosterstern (U1), Hallerstrasse (U1) in the North, and Dammtor (S11, S21, S31) or Hauptbahnhof (all lines) in the South.
If you want to rent a canoe or a small sailing boat, here are some rentals around the lake: Barca Hamburg, close to metro station Hauptbahnhof (all trains), An der Alster 67a: http://www.barca-hamburg.de Besides the sailing boat and kayak rental Barca is a nice coffee shop, bar and snack shack with spectacular views over the lake.
Bobby Reich, close to metro station Klosterstern (U1), Fernsicht 2: http://www.bobbyreich.de/netscape.html This kayak rental also belongs to a café and restaurant but I only recommend to have drinks here, e.g. outside on the pontoon.
Bodo’s Bootsteg, close to metro station Hallerstrasse (U1), Harvestehuder Weg 1b: http://bodosbootssteg.de/. This kayak rental also has a small coffee shop onsite.
Biking at Lake Alster If you want to bike the lake the cheapest and easiest way is “StadtRad”. This rental has rental terminals spread all over the city. You only have to sign up online for EUR 5, which will be refunded with your first rental. At the bike terminals you identify yourself with your debit or customer card and receive the code for the bike lock. More info is available under: http://stadtrad.hamburg.de/kundenbuchung/process.php?proc=index&f=511 Just be careful since German bikers can be very rude during rush hours!
ELBE RIVER If you want to explore Elbe River and the harbor area you can walk or bike along the riverbanks starting at the new HafenCity (close to metro station Messberg (U1) and Baumwall (U3) or Landungsbrücken (U3). The promenade takes you up to Övelgönne with its old museum harbor and even further to Blankenese. This is a total of up to 8 miles (13km) per direction.
Another nice walk or bike path goes from Strandweg in Blankenese (close to metro station Blankenese (S1, S11)) up to the nature reserve Wittenberger Elbwiesen in Rissen. This path is 2.5 miles (4 km) long per direction.