Hi Joris,

Starting out with landbased fishing is not easy and also Wellington is not the easiest place to fish.
You could get yourself a kayak and paddle out a few hundred meters into some of the sheltered bays.
That would increase your chances of catching something dramatically.

What have you tried so far and what kind of gear do you have?
Surfcasting can be quite good but requires a bit of patience, knowledge of the beach and the right bait. When landbased and fishing so close to shore your best times would be in the evening just before sunset.
Try to find a day when the major or minor times are also around sunset to increase your chances.

Bait is also very important. If you're surfcasting off a beach try mussels or crabs or even store bought shrimps as bait.

If you're fishing in slightly deeper water off some rocks or around headlands, try to use berley to attract fish. 

If you give me some pointers of what you have tried so far, I might be able to give a few more tips.

cheers
Mark