PFD - It doesn’t need to be top of the line, but make sure it’s comfortable enough to wear it won’t do any good. For years, I was pretty bad about wearing my PFD. Not any more - I have a comfortable vest that I keep on at all times.
Paddle – You have to propel that kayak somehow. As anyone who has slipped a paddle overboard can tell you, hand paddling is no fun. Seriously, test a few paddles before making a choice.
Whistle/Airhorn – Required safety equipment. Keep an audible signaling device attached to your PFD so it’s with you in the event you become separated from your boat. I keep a whistle on my vest and an airhorn in the boat.
Waterproof Flashlight (preferably a headlamp) – If the unexpected occurs and day becomes night, a good light allows you to signal and warn other vessels of your presence, as well as navigate at close range. A white light that may be displayed in time to prevent a collision is required safety equipment at night. If you are intentionally staying out after dark, a lightpole that stays on at all times is essential.
First Aid/Survival Kit – A basic kit in a waterproof container could end up saving a day on the water, or more. It’s light and easy to stow, so why not carry one?
Waterproof Storage – Everyone ends up in the water at some point, so it’s best to have a safe place to keep your wallet, keys, phone, etc. If your boat doesn’t have a wallet hatch, a drybag is great. Plastic food containers work, too.
Fishing License/Launch Permit – I keep mine in my wallet or pinned to my vest, depending on where I’m fishing. Always pay attention to local regulations.
Float Plan – Always let someone know where you’re going, and when you should be expected to return.
Water/Food – I make sure I’m prepared with “fuel” for the inadvertently extended trip.
Proper Attire – Matched to conditions: waders/drygear, sunglasses, neck gaiter, hat, etc. Obvious, but poor planning in this department can destroy a trip.
Here are various other items that aren’t “true essentials” but are great to have:
Lightpole and Flag – See above. Visibility is key: the flag for daytime, the light for night.
Waterproof VHF Radio – Cell phones don’t work everywhere, and it’s nice to have a direct line to emergency responders. If you’re bored, you can also use it to chat with fishing buddies or gather intel, but please don’t jump on CH68 and rat out the guy who’s hammering fish three pilings down.
Knife – This isn’t optional if you’re using an anchor, as you may become stuck and need to cut the anchor line in a hurry to prevent swamping. I like a knife that clips to the outside of my vest.
Rod/Paddle Leashes – Secure your stuff, or you’ll likely eventually wind up sacrificing some expensive gear to the depths.
Signal Mirror – In an emergency, it may become necessary to flag down help. I use an old CD, just in case I need to get someone’s attention in a hurry.
Crate/Gear Storage – Most people use some sort of storage crate or box in the tankwell (I’m one of those weirdos who does not, though I have in the past). Setting up a crate with rod holders, camera mounts, etc. could be an independent article.
Anchor/Stakeout Pole/Drag Chain/Drift Sock – Means of controlling your boat on the water to give you the best shot at fish, depending on conditions.
Measuring Device – This is an essential if you plan on keeping fish that must be over a minimum size. Plenty of manufacturers offer trough devices, or you can just use a yardstick or sewing ruler.
Fish Bag/Cooler/Stringer – Some means to keep your catch relatively cool and fresh. Most of my fish are small, so I typically use a commercial cooler bag, loaded with ice.
Fishfinder/GPS – Two devices (or a combo unit) that can be a tremendous boon to your fishing.
Spare Rope/Bungee – Hundreds of uses, from tying off on structure to towing an exhausted or injured paddler.
A Friend – The buddy system is good practice. Plus, you’ve got someone to corroborate your fish stories (or tell everyone the unfortunate truth, i.e., “He got skunked.”).