Hiking highlights around Wellington: Red Rocks (NOT the amphitheater in Colorado), Paekakariki, Mt Victoria, Colonial Knob, Mt McKerrow, etc.
Red Rocks has seals which are a sight to see! It's also a site of filming for the Lord of the Rings, fun fact: when Frodo, Sam, and Gollum reach the Black Gates and Sam falls down the rocky cliffs thing.
Paekakariki is drop-dead gorgeous. Mind-boggling, even. It stretches along the coast and has a few suspension bridges. Be warned though, it's a one-way track from one train stop to another and there are a lot of stairs... I would recommend starting at the north end of the trail because then more of the stairs are going downhill.
Mt Vic is in Wellington and pretty short, but also beautiful (like all of New Zealand) and is also a LoTR film site: the road out of the Shire when they run into the Black Rider.
Colonial Knob is pretty easy and through farm country, so lots of cows and sheep!
Mt Mckerrow is where we went backpacking one weekend. There are several tracks in that area and the Mckerrow track leads to the Clay Ridge Track which led to the campsite we were aiming for. These are 'advanced' tracks though (Department of Conservation rating) which means they're pretty steep and you have to pay attention to where the trail is going.
Budget hacks: As I said before, the farmers' markets are the move for produce. Very affordable, but bring your own bags and cash!
If you're into Taco Tuesdays, Rogue and Vagabond has those sweet $2 tacos.
Koha is great because it's essentially pay-what-you-can! Koha means gift in Te Reo Maori, so when you leave you're encouraged to leave a donation. The people who work there are super nice and accommodating. However, they mostly do just coffee, hot chocolate, and tea--nothing fancy like lattes and cappuccinos because they don't have the equipment. Sometimes they do have yummy baked goods though!
The Saturday night markets have a truck that does two sizable samosas for $6.
There are two supermarkets, mainly: Countdown and New World. I've found Countdown to be more affordable in general.
Also if you are 18 (which you probably are) and want a bargain, JJ's on Cuba St has $10 jugs and free pool/billiards upstairs (free if you buy a drink, so mostly free).
Also a bunch of good thrift shops around. We found a 'tour de thrift' of shops in the Wellington area.
Favorite things to do: Botanical Gardens (Botany). Beautiful and free. They're a great place to walk around or sit and read or chat or what have you and they're super close to campus. Plus there are GLOWWORMS here (so you don't have to pay for the caves if you don't want to! You just have to venture and find them).
Te Papa Museum: Also free (except for like one exhibit) and very fun and informative to wander around.
Watching the sunset on the water: Stunning. Need I say more?
Oriental Bay: Great place to hang if you want to go to a beach/pier area.
Mt. Vic: Great panoramic views of the city.
Cooking with friends/Movies and Wine nights: Great, low-budget way to spend time with the homies.
But, you know, it's your life so live it the way you want I'm not here to tell you what to do.