This is an easy bike ride along a signposted route which tunnels under the airport then wends its way through quiet suburban streets to the giant Zephyrometer (wind-wand sculpture) at Evans Bay. From here it follows the coastline over a dedicated cycle path into the central city. In Wellington you'll find several hostels of reasonable quality and reasonable prices, including an excellent Youth Hostel.
Wellington is the departure point for ferries to the south island, but for those staying a while the city has much to offer. Popular activities include Te Papa (the national museum), the cable car, theatre, art galleries, parliament, Weta Workshop, Zealandia wildlife sanctuary, ...
Known colloquially as "Windy Wellington" the city is often buffeted by strong winds, funnelled through the Cook Strait by surrounding mountain ranges. But on a good day Wellington is hard to beat. One great thing about this city is that it is quite compact. Hemmed in as it is by the surrounding hills and harbour, urban sprawl has been unable to take hold. Consequently you can visit Te Papa, take swim at Oriental Bay, dine at one of the many restaurants (according to Lonely Planet there are "more bars, cafés and restaurants per capita than New York") and then go out for a movie or show - all within easy walking distance.
If you have time to explore Wellington then this is an easy, mostly flat ride and makes great day trip, passing through plenty of swimming bays, with cafés dotted around the coast. The round-the-bays loop is popular with Wellingtonians and if the weather is fine you're bound encounter locals out for a lunchtime jaunt or perhaps taking the long way home.
The great thing about this route is that - even if you're a stranger to the city - it's virtually impossible to get lost. Just keep following the coastline until Owhiro Bay, where you turn inland and cycle up the valley through Brooklyn and back down to the city. If you're keen, you can continue around the coast from Owhiro Bay for about 5 km along a 4WD track which takes you to a seal colony at Red Rocks.