Seeing you are on $50 a week there is lots of free stuff to do in Wellington. The tour of parliament is free and goes every half hour. Also the mueseum of Wellington city and sea is also free. It outlines the history of Wellington and in particular its nautical history. Finally as mentioned earlier there is Te Papa which being a global traveller is the best mueseum I have ever seen (I'm not just saying that because I'm a Wellingtonian) Most of the displays are free but if you want to see the Lord of the Rings Exhibition it will cost you $12. You are actually going to be there on an excellent weekend as the food show is in town and it highlights New Zealand food. It is a the Westpac Stadium from Friday 12, Saturday 13 and Sunday14. There is also The Big Look See next weekend where the arts are having a whole lot of free activities. The two professional theatres are having open days so you can see behind the theatre and preview their new productions and at Bats theatre you could be in play. Through the following link you can access the activity planner
http://www.wellingtonnz.com/Events/The+Big+Look-See.htmAlso take a walk along the waterfront. There is a lot to see including a writers walk which includes quotes from famous Wellington authors.
Some things to do that cost money.
take the cable car ($3 return) to look around the botanical gardens
Go to the Karori Wildlife Sanctuary to learn about New Zealand Wildlife and the early Wellington gold rushes. This costs about $8 but is fasinating.
A wellington rover tour has lots of tours if you interested
http://www.wellingtonrover.co.nz/.
Visit Wairarapa either through a tour or Tranz Metro wine tour which is excellent value as for $95 you get transport to Martinborough for the wine tasting which is a bit cheaper than the tours.
http://www.tranzmetro.co.nz/SpecialTickets.aspx#waipacAs mention Cuba Street is very funky. For drinking and eating try either Matterhorn or Good luck bar. Indigo and bodega are excellent for local music and visiting artists.
My favourite activity is doing the coastal drive around the many bays that make up Wellington. However, to do this you will have to hire a car. Make sure you stop at the Massey Memorial. If you do you can stand on the hump at the end and face the columns. Then sing the acoustics are amazing. At scorching bay stop off at the Chocolate Fish cafe, my favourite cafe in wellington.
Take the ferry across to Matiu/Soames Island. It use to be a quantine centre and has a fasinating history or stay on the ferry and go to Days Bay across the harbour
http://www.eastbywest.co.nz/Wellington is also the theatre capital of New Zealand. An excellent New Zealand play is currently on at Circa. Check it out.
http://www.circa.co.nz/whatson/studio.htmFinally hire a kayak and kayak around the harbour.
http://www.fergskayaks.co.nz/Am happy to offer my services as tour guide although be warned I am called the tourist nazi by my friends!!!