When staying in Wellington, there are a lot of walks to do everywhere. You can go to parks, reserves, walk on beaches or in the bush.
You can stare at the beauty of a 800 years old Rimu tree in Otari-Wilson's Bush. You can also enjoy the views of the many difference bays along the Miramar Peninsula. There are many pleasant walks with amazing views in the parks along the shore.
If you feel like escaping the city for a day or two, you can push it to Rimutaka Forest or even Tararua Forest, It is a maximum two hour drive along state highway 2 in the direction of Masterton. There are many small and loop walks as well as greater walks that can take you 2-3 days where you can stop and sleep in the huts along the trails where you can meet other trampers.