I Freedive spearfish and will be listing some of the essentials needed. I'm sure that a few of you scuba dive, so feel free to add on.
Mask: Obviously very important. Most tend to buy low-volume masks with a black skirt so that light doesn’t penetrate your field of vision. These masks significantly reduce glare. Here is an example:

Snorkel: In my opinion, a snorkel is a snorkel but I have heard that the ZOOM snorkel from Omer has one of the best mouthpieces and is made of a high quality, yet flexible plastic.

Fins: How do you get to the bottom quickly? You need long fins. Fins come in different stiffness’. Cressi Gara 2000’s are stiffer than the Gara 3000’s. There are also Carbon fins that are extremely durable. Normally Closed foot pocket fins are used by freedivers and open foot pocket fins are used by scuba.

Speargun: There are some big, bad guns out there. But I’m going to talk about “affordable” guns.
Spearguns fall into two main categories: Band Guns and pneumatics. There are also the most basic spearfishing weapons such as polespears and Hawaiian slings.
Band guns can be described as “a gun that converts potential energy to kinetic energy trough the use of tension.” The bottom line is that the actual spear has notches on its surface where the metal wishbone of the band gets pulled back to. Some “affordable” Brands are A.B. Biller, Sea Hornet, JBL.

Pneumatic Guns rely on compressed air for propulsion. The expanded gas is kept and can be re-compressed underwater. Some Examples of pneumatic guns are Cressi-Sub, Omer, Mares.

Pole Spears go way back and like I mentioned are the most basic spearfishing weapon. They function like a billiard cue with a rubber band attached at the base. The band gets placed in between your thumb and forefinger and you stretch your arm close to the tip of the spear and when you see the fish…well, you know what comes next.

Hawaiian Slings are similar to pole spears in that they are basic in design. A Hawaiian sling is like a sling shot, when the fisherman carries a small tube with a band at the end. The band has a cup at the end where the spear butt fits and gets pulled back.

Note: Pole Spears and Hawaiian slings are a great way to get into the sport and are the only weapon that you can use in the Bahamas and other Caribbean islands.
DiveFlag – A diveflag is VITAL AND REQUIRED when diving. It tells boaters that they have to stay at a minimum of a 300 ft radius away. I have had close calls myself in Miami where jetskiers think that a diveflag is a speed bump. ALWAYS have a flag.
Knife – You can never go wrong with a knife. Most dive knives have a line cutter incase you get tangled. Knives are also used to dispatch fish.

Wetsuit/Rashguard/Diveskin – It does get a little chilly here in winter so a wetsuit around 3-5mm won’t hurt.

Rashguards are shirts and/or shorts made out of lycra, a brand name of spandex. They fit snug to the body and WILL attract beach views gaze

Diveskins are full body rashguards.
Not only do Rashguards and dive skins look stylish but they protect against UV rays and jellyfish.
I just threw together some basics, I am sure that I’m missing a few things so I will amend the list as needed.
Dave