The city is fantastic. People are really nice, lots of culture and I like way they mixed industrial with cultural - and posh with normal. The recent upgrade of the old docks made it into tourist attraction #1, with the € 700.000.000 (once completed - the Sagrada Familia of Hamburg....) Oper as 'highlight'. The 'beaches' in Blankenese next to the Elbe and the surrounding hillsides are nice for a picnic.
Not much rock close by. Might want to visit the Ith. Some really good climbing in the Selter. Check 'Hoch im Norden' by Panico.de.
There are several climbing gyms. Nordwandhalle is the biggest. There also a bouldering gym: Salon du Bloc. Even got a communal climbing wall on an old bunker. Went climbing there on a sunday with a few kids and appreciated the place - not because of the climbing. It's located in part of town that has become hipster territory - yek -, but the bars, food, music in the streets, mixed with drug-dealers and alcoholics make it a nice place to be. Most impressive thing about hHmburg(ers): everybody is very, very very polite.
Would be a fine place to live - except for the lack of rock.