New to free diving and living in Sydney, but moving to Hamilton. 

5mm is a good choice for an all-year suit, make sure it’s open cell and fits well. Ask to try some suits before you buy, I always eyed up a certain brand till I was in the market for a new one and used the opportunity to try it. Figured it didn’t fit me at all so I went with the brand I was used to which fits like a glove. Don’t skimp on it either - the soft, stretchy and comfortable suits are a tad more expensive and not as forgiving when rock hopping, but are well worth it when you take care of them. Rinse in cold water after use, hang in the shade using a wide hanger and well - I use waist high pants and have cut the suspenders of my farmer johns for reasons. If you cut the suspenders reasonably high, you can flip the top part over and create a waterproof seal.
I’d spend most of my budget in that order:
suit - mask - gun - knife - fins - weight system - snorkel - gloves - booties - float/line

In terms of suit quality, pretty much every suit on the market is produced by Sheico - which in this case IMO is a good thing, since they’re creating a high quality product. If it states “made in Thailand” go for it.

Also, if you get a good open cell wetsuit then you may find that a 3mm will do you for many of the warmer months and a 5mm for the cooler months (if your budget can stretch to two suits).