Taxi to HI is probably $20-30 (quite high for short distance due to AKL airport gouging). AFAIK the shuttle bus for HI needs to be pre-arranged (ie it isn't a normal shuttle).

SYD-AKL-RAR with through checked bags (show your onward ticket at SYD and NZ will through-check) does not require clearing of immigration or customs. You will have transit security (so don't bother with large liquids at SYD as they will be confiscated at AKL). If your arrival gate is on the main pier (or remote stand) then transfer security is on the right at the base of the pier, if your arrival gate is on pier B then you walk past immigration and through the 2 duty free shops and the entrance is roughly straight ahead.

2:30 is more than ample. If you don't already have a boarding pass you can collect it from the transfer desk above gate 4, or from the lounge if you have lounge access. From the transfer security exit, transfer desk is to the right, up a level and past the shops on the top level (and then on your left), whereas the entrance to the lounges are to the left and then down a corridor also on the left.

If AKL-CHC is on a separate ticket on NZ then you are already checked in (at time of booking) and only need to get baggage tags at the kiosks for checked luggage. If no checked luggage you can go directly to the gate or lounge. If not on separate ticket then you may also need to check in, using either the kiosk or check in counter (warning this can be slow especially first thing in the morning). Bags are dropped by the passenger onto baggage belt then through security. Most of the time arriving 30 minutes before departure is ample, but 45 minutes gives a buffer for extra busy times (check in queue if unable to use the kiosk, &/or security queues).