We live in Island Bay and I would argue the best fish and chip shop is Seaview Takeaway at Lyall Bay. Bigger beach there and also THE best cafe in Wellington - Maranui. Tuesday is a good day to visit as they do a full breakfast and coffee for $16
I would definitely explore some suburbs though to "see" Wellington as a place to live.
Walk along the waterfront too. Great on a sunny day. A trip up Mount Vic is also worth it for the view of the city.