Basic equipment needed?

I'm looking into getting into kayak diving. I was going to do when I lived in Florida but regulations were really tough and never did.

What gear do you guys use?
What thinkness suit?
What brand gun? Etc

Thanks for any help !

Spearfishing?  What's your experience level with tank/free diving in general?

Here's a short list:

Fishing license

Sit on top kayak, 14' or more
Paddle
Kelp clip
Anchor

7mm open cell wetsuit
60-80cm open muzzle speargun, 70 is standard
Float line or speargun reel
Well fitting mask, preferably low volume
Marseillaise weight belt
~ 20 lbs of pass-through lead weights in 2lb increments
Neoprene gloves, 2.5mm at a minimum
Neoprene booties, 4mm at a minimum
SMALL dive knife
Closed heel fins that fit over booties
Underwater flashlight, 500 lumen or more
Method of timing dives

Some of those items are pretty specific, but there's a functional reason.

Sit on top kayaks are necessary so you can get on/off, and they need to be big for stability, weight capacity, and long paddles.  Seriously consider a Trident 15, Tarpon 160, or Scupper Pro TW.  Make sure you have enough access to the hull to store everything but you and your paddle.  You'll need a kelp clip or anchor to keep your kayak from floating away so you don't have to buy a new one every time you go out.

Wetsuits need to be bare rubber/open cell inside for warmth and flexibility.  You could use a 7mm lined scuba suit, but you'll end up buying an open cell eventually if you're freediving.  I'm dumbfounded why scuba divers don't use them, myself.  I do when I use tanks, but a more crush resistant neoprene is preferable for scuba use.  You won't regret going the custom route right off the bat.

Euro/pipe spearguns are the standard.  You could buy one Rob Allen 700/70cm and never buy another speargun for Northern California.

You'll need a float line or reel for those times you shoot a fish in a cave and the shaft gets stuck.  Assuming you're freediving, you don't want to leave your gun on the bottom.  You'll lose it.  If you're tank diving you may want to try a pole spear for the simplicity.  Lots of line in the water can be a hassle on tanks.

Low volume masks require less air from your lungs to fill them as you dive deeper.

Rubber belts are necessary as your body and wetsuit compress the deeper you dive.  Otherwise your weight belt will start rotating on you or be uncomfortably tight.  I don't know why scuba divers don't use these either.  A crotch strap is a great idea too.  Very few people use them for some reason.  They allow you to keep the belt looser, and can be made from $2.00 of nylon webbing and clips from the hardware store.

Weights need to be pass through to allow the rubber belt to stretch properly.  See "bullet weights", "tin man" weights, or neptonics weights for reference.

Gloves and booties need to be thin enough to keep dexterity, but thick enough to keep you warm.  4mm is my upper limit with both.  Maybe thicker on scuba.

Small dive knife to cut you from an entanglement, or to finish off fish.  Needs to be kept somewhere out of the way, or else it will catch on kelp.  Legs and arms are a no-no.  Preferably in a thigh pocket or on your belt.

You'll need a flashlight for finding and shooting fish in caves.  Anything under 500 lumens can be useless at times.  I use one with 1100, the brightest on the market right now.

You may want a stringer to keep fish on your belt, so you don't keep swimming back and forth to your kayak.  They're essentially a spike connected to a tuna clip with 500lb monofilament.  Don't attach one of those metal hoop stringers to your body unless you're on tanks.

You'll also need some way to time your dives so you stay safe, in addition to your freedive class.  Aeris F10 seems to be the standard.  Oceanic GEO 2.0 is the same watch but with scuba and nitrox capabilities. 

Jeremy gave you a lot of suggestions and things to think about. 

If money is not a problem go out and get the best in all categories and go take a Freedive class to learn how to use it all properly.  If you want to do this on a budget you can always buy the items he listed that are more reasonably priced.  Then as you learn what you like and don't like you can replace items and if the first purchased items are still good you can use them as a back up or loan them out when a dive buddy forgets something.  Just like any hobby or sport extra equipment is always nice to have. 

I always bring an extra set of gloves and booties and leave in the truck in case a pair rips while diving or putting on.  I have a 3mm and a 5 mm set of gloves that I change out depending on how cold the water is.  3 mil is definitely more flexible but a 5 will break in and be flexible enough and keep you warmer if that's what you like. 

New divers usually take awhile to find the right snorkel/mask set up.  When you find the one that fits perfect buy a second one.  I have lost a mask and had another masks band break.  I would have had to call a dive if I didn't have a second one. 

Some weight belts fit different sized people better.  If you are only 110 lbs then 20lbs of weight will sink you.  Learn about buoyancy and don't over do it.  I have multiple belts and like the stiffer rubber ones that won't slide down as easy.

Having a dive flag is a great idea if you dive around boats or want other divers or people from shore to see you.  It really is hard to see you when waves increase. 

If you are already kayaking then you probably have a VHF radio and fish finder but if not a VHF is a must and it sure helps to have a fish finder. 

You will have to figure out the stringer thing on your own.  I don't like attaching fish to my body but many have no problem with it.

A knife is a must.  Just keep it sharp and be able to access it with either hand in case one gets tangled and you can't reach it.  They rust and are easily lost so don't spend to much on one. 

If you are a new diver the flash light lumens won't make that much of a difference because you probably won't be able to hold your breath long enough to use it.  But when you get good a bright light makes all the difference.  I won't use anything less than 1000 lumens.  Some don't need something that powerful but I do.  I miss fish with a light that's half that bright.

As for wetsuits get a 7mm open cell like Jeremy said as you won't regret it.  You will use a water/conditioner combo to spray on the inside so you can slip into it.  You can find these on sale for approx $200 but they do go up to $500 or more for off the shelf and custom suits.  Unless you have weird miss shaped body parts an off the shelf one should be fine.  If you have a shop close by try one on.  If you feel to restricted in your chest or your neck then try a larger size.  I had one that was so tight around the neck that when I moved my neck I think my carotid artery was being pressed and it felt like I was going to pass out.