Pier is probably better for catching bait (mullet/kahawai), just buy one of those sabiki traces from any tackle store and use small peices of bait (bonito or sheep/chicken/ox heart etc). You’ll catch mullet and kahawai at lyttelton but I’m pretty sure there’s no wharfs you can fish off, only off the breakwater. If fishing off the breakwater you’ll probably catch more spotties than anything else. Pick a spot with no weed and wind in fast without stopping to prevent getting snagged.
As for the Avon I don’t know, I think they go pretty far up. I just fish by the boat place at new brighton and have good luck sometimes, just have to wait for a school to come past.