See "standard warning" at end re the inadvisability of snorkelling alone.
It depends on what you want to see and do, but a brief answer is
"almost anywhere that you see sea-water.
Auckland is not renowned for its water clarity, but it can still be fun (speaking from personal experience). You'd want to stay away from the sewage ponds at Mangere, and some of the major creeks/streams that drain storm water from industrial areas.
The Auckland East Coast beaches are all relatively safe. Rock areas between beaches are not usually anywhere as violently dangerous as the West Coast beaches but have numerous rock pools and relatively low depth areas with a reasonable amount of sea life.
I've never done it, but the old explosives wharf - now just Map - Orakei Wharf - zoom out to see where you are - would be fun foir snorkelling. There MAY be a "no swimming" requirement - but probably not. I've fished there, but never swum.
Map - Great Barrier Island is an utter gem. Several hour ferry ride so not zero $ but it's a completely different world. Beaches and hidden spots with few people (and a few very strange ones :-) ).
An utterly superb summer venue (good also in winter but colder) is "Goat Island Bay" a few hours by bus North of Auckland, but worth the trip. Transport may require some ingenuity - but to get a bus most of the way for $NZ1 see below "The Naked Bus".
"Goat Island Bay" is the site of a university marine research station and also a marine reserve where all marine wildlife is protected. I snorkelled there some years ago and have never seen so many fish anywhere else in NZ. This may not always be what happens and things may have changed in more recent years, but you can inquire re the latest situation.
**Goat Island Bay Marine Reserve images
Lots of information here
The Naked Bus Company
The Naked Bus Company provides a bus service which covers much of New Zealand (both main islands) and which can cost as little as $1 full fare IF you book far enough ahead. As the date gets closer prices rise, but are usually still reasonable. A ride to Warkworth (near GIB) in March is still $1 on some days at some times. On March 8th you can catch an Auckland To Wellington bus for $9.99 - IF you book now. Interestingly - on May 8th it's $19.99
Naked sleep {wazzat?} - cheap accommodation
Standard warning - which will be ignored:
There are no places anywhere that are good for snorkeling by yourself if you want to live to a ripe old age. Snorkeling is at the very benign end of hazardous underwater pursuits. While it's about as safe as any such activity, it's not wholly safe. But, you know that.
I had a friend who frequently engaged in a 'safe' water sport by himself in Auckland. He died. He was very competent and very experienced and everyone was extremely surprised. But, he's still dead. Nobody knows exactly how it happened, but there is a good chance that if he had had a "buddy" present that he would have lived.
Auckland map:
Purple - best likelihood of calm and somewhat clear conditions.
Waitemata harbour (East/right) is bearable for swimming. Sometimes Orca come in past the harbour bridge ! Rare.
West Coast beaches (red ovals) are GREAT fun but great care needed. Careless people die.
Muriwai - long ocean beach with breakers. Rocks to south of road access. People fish from here and on occasion get swept off and die. A competent snorkeller WITH FINS could have great fun there. I would only do it with wetsuit and fins !!!
Piha . Bus from Auckland. Great surf beach. Some interesting rock pools, blow holes. You can swim around Lion Rock. Surf can be large. Getting back in through breakers cvan be demanding (ask me how I know :-) ).
Karekare - South of Piha (no circle). Bus maybe.
Rangitoto Island - Ferry from Ak. Few people. Rugged lava coastline. Water as good as anywhere here.
Waiheke Island. Ferry. Some lovely beaches, lots of rocky bays, headlands , ... .
LARGER VERSION HERE
