For visitors to Wellington, this page attempts to provide some idea of how to get around.
To begin with, I will point out two online maps:
Wellington is a fairly compact city. As such, if you are relatively fit, and the weather is fine, there is no reason why you could not walk everywhere. The difficulty, however, is that, while the central business district is flat, Wellington is a hilly city. In particular, the University is uphill from all the recommended accommodation options (although there is only a short climb from Weir House). If, therefore, you (for whatever reason) would prefer not to walk, you have several options:
Wellington's Bus Service is provided by Stagecoach. The Wellington City Council has a reasonable website for timetables and routes. Bus routes passing the university are: The 22 / 23 to Mairangi, the 20 to Highbury and the 17 to Karori Park (although the 17 only runs on weekdays, and infrequently outside the main term time).
All these busses travel west along Courtenay Place, then North along Willis Street and Lambton Quay, and then South along The Terrace. (note that in the PDF map above, North is on the right) See here for a map of bus routes in Wellington central.
Price will be $2 from anywhere in town (between Courtenay Place and Lambton Quay) or $1 from The Terrace.
A Cable Car runs from near the South end of Lambton Quay up the hill to the Botanical Gardens. You can get off either at Kelburn Park, near Weir House and walk up to the university (this is signposted "University" at the stop), or get off at the Botanical Gardens and walk down. The Botanical Gardens stop offers nice views over Wellington and the Harbour on good days.
Fares are $1.80 each way for adults, or $1 for students (you may be asked for ID). The Cable Car is marked on both maps above.
This webpage author has never caught a taxi to the university, so he can't comment on potential fares. Perhaps $5 - $10. You could check the webpage for Wellington Combined Taxis for details.
Note that you are not automatically expected to tip the driver in New Zealand (the same applies for eating in restaurants, et cetera).