This is primarily going to be about getting around in Wellington. First off you can walk to most places provided you do not have any health issues that prevent you from doing so for long periods of time. The only down side to walking is it takes a lot longer than just hopping on a bus or taxi. Their bus system is fairly easy to use though and really not too expensive if you keep some things in mind.
The bus from the airport to the bus station in Wellington is about 8 bucks. If you plan on seeing the city the day you arrive ask the driver for a day pass. It's 9 NZD and you can use it all day and it really saved a lot of money if you look at most stops cost about 5 and if you want to go to more than just one place that quickly adds up.
Also be aware that Wellington is divided into 14 zones. (Maybe other cities do this as well but we walked everywhere in Queenstown so I hadn't encountered it) The more zones you drive through to get to your destination the more the cost. http://www.metlink.org.nz/fare-zones/
Pretty much what we would do if we had time was walk as far as we could and catch the nearest bus in the last zone so our fare was cheap. Normally you'd just do as I said before and get a day pass...but it was the end of the evening and the 9 dollars was more than 3 the trip really required.
Plan for your bus to break down. Especially if time is of the essence. I say this because at least 3 different times the bus I was on jumped off the power lines and we had to stop and catch the next bus. The wait was never too
long but if you're late for your flight you might be screwed.
You can actually save money getting back to the airport if you take one bus near Miramar. There's a stop that's quite close to the airport and it cost 5NZD from the station. Yes you'll have to walk about a mile (though you can clearly see the airport when you get off) but again if you wanna save money or are nearly out of cash then this is your best bet. Again not the best option if you need to hurry and catch your flight, but we had time that day and not a lot of money to do much else. The airport flyer is not expensive though and probably more reliable than the buses so really it's all dependent on how much you need to save.