In my opinion the Best spot close to Timaru for Surfcasting is St Andrews Opposite the Golf course. Good spot for Elephant fish from late October through to February. However this spot is also notorious for Sevengill sharks so if you dont want to catch sharks it may not be for you, but to be honest all beaches down here have sharks so whether you target them or not they are going to be part of your catch, Best way to lessen your catch of sharks is to use shellfish or crab baits.
All the beaches from Birdlings flat right down to makikihi Have pretty much the same species, Rig,Elephant fish, Shark, etc.
For Moki you could try around Scarborough or some of the rocks around the Timaru breakwater but there are heaps of snags off the breakwater so if you fish there be prepared to loose some gear.