Did a bit of that when I was a kid in SoCal skin diving around the kelp beds off Palos Verdes.
Best to go out with someone who's done it. The sling takes some practice as does getting close enough to the fish to use it. Check with the local dive shops for info on equipment and best locations. But as was mentioned, try borrowing the spearfishing gear first. Some kind of protective gloves might be a good idea. We never used them; but that was a long time ago, and we were young A sharp dive knife might come in handy. Also, check local/state fishing regs.
Try not to have your bleeding catch tied to you for very long while diving.
On flat days, I paddle out with a longboard along the reefs near shore. Tied to the longboard leash is a boogie board. Strapped to the top of the boogie board with a couple of bungee cords is a rubber maid bin. This way once you shoot a fish, you can throw it right in the bin, keeping most of the blood out of the water. It has to be flat, because if the boogie board flips you will lose your fish ( and ice), unless you take the extra step of having them in a mesh net inside the bin. Bring a knife, just in case you get caught up in mono line or whatever. Bridges and jettys hold the most fish, but it is illegal to shoot fish near them here in Florida, but every state has their own laws. Be safe, have fun. Watch out for boats, jet skis, and your buddy's spear.
Check out spear board.com. there's a mid Atlantic section. Its also dangerous, especially going alone. Research shallow water blackout. Ive been wanting to get into it but right now its spend $400 on equipment or put it towards a new board which I desperately need.
I would dive without a spear first to learn how to free dive including relaxation/breathing techniques, equalization, and figure out your proper weight. Proper weighting yourself is essential to free diving.
After you get more comfortable take out a sling, then try a spear gun. I dove with just a camera for about two years before I starting shooting fish.
As mentioned can be very dangerous so go with a buddy.