I have seen many recommendations - many different recommendations - and I must admit that I personally hesitate to recommend any particular type, make or model of kayak.
Most fishers, myself included, have chosen a so called sit-on-top model, that facilitates getting in and out, but 'real' kayaks of the arctic/greenlandic design with a deck and a hole are just as well suited. The drawback is a bit more difficulty in entering and leaving, while the advantage is superior speed in many of the closed models.
But still, judging from books and magazines the open ones seem to be preferred by most fishers. I would not exclude one for the other, but give both designs a try before I bought. I chose the open one because my kayak is not only a fishing vessel, but also a means of transportation, and I want to be able to get in and out with no hassle... and with my waders and large wading boots on.
No matter what you do, make sure you get a stable model. These are typically slower and a bit harder to paddle, but nothing that will have influence on your fishing. Stability will! The more stable, the more maneuvers you can do unpunished. Go for stable hull shapes and wide kayaks.
Some kayaks are or can be fitted with a rudder. That is worth considering. Kayaks float high and tend to drift and turn in the wind. A rudder, which is operated with your feet, is a great help in keeping position and direction.
I will not recommend specific brands or models either, because there are so many out there and I have tried so few. The only things I will recommend is aiming for light weight and easy maintenance. Most modern kayaks are made from some kind of polycarbonate, which is both light and very sturdy. Kayaks can also be made in wood or glass fiber, which is heavy, and more exotic materials such as carbon fiber, which is expensive, or fabric over a frame, which is indeed charming (and lightweight), but also more vulnerable than any of the other materials.
Folding kayaks are an option if you need to pack really compact, but not in itself a benefit to your use of the vessel.
Other kayak gearPaddle
You need a paddle. Take what the dealer recommends within your price range. You might consider a two piece paddle for easier transportation, but a one piece has offered me little hassle over the years.
A personal flotation device is a must if you want to travel or fish over deeper, colder water. You WILL roll over! Be prepared. A tight fitting sailor's vest or small self inflatable vest is the right choice. Make sure it does what it is supposed to. Try it in warm water. The best ones have radar reflectors, a whistle and other stuff that you will appreciate when you are in. Do not go cheap here! See the chapter on rolling over below.
If you fish warm water only, and you are an able swimmer, the life vest is not necessarily a must.
I will cover more security equipment in the section on security.
Some kayaks come without a soft seat. Judge for yourself whether you need one or not, but I find my soft seat very comfortable. It offers soft cushioning and a backrest plus a large pocket that usually holds my thermos, drinking bottle and lunch pack. Seats are fitted in different ways. Mine detaches for washing and drying and I usually stove it indoors.
You can buy kayak clothing, which will set you back hundreds of dollars. Specially designed wet suits, breathable jackets, soft shoes and what-have-you. If you fish cold water and have the money, go for it. I personally use my usual fishing garments: neoprene or breathable waders, wading boots and fishing jacket. It works fine!
If you fish deep and in waves, get something that fits tight. Loose, breathable waders will fill with water if you go in. Tight neoprenes will help you float and not fill up with lots of water. And do not believe anybody who says the waders and sailing do not combine! You will not turn upside down, but float nicely with your legs in the surface.
I have always brought an anchor. It is convenient and keeps you in place in wind and current. It is also a safeguard. If your kayak rolls over the anchor drops to the bottom, and the kayak is fixed. A small ½ kilo or 1 lbs. anchor will hold the kayak. A small piece of chain just above the anchor will aide its grip.
Keeping stuff dry is important. Clothes, food, gear. Dry bags are available in all sizes and shapes. After having rolled over with a 500 dollar cellular phone in my pocket, I now even have a small one for the phone! Cameras, phones and the like does not take a dunk in the water lightly.
Whether you intend to fish from the kayak or just use it as transportation, rod holders of some kind is necessary. You can basically select two strategies: vertical or horizontal.
Horizontal usually means simply strapping the rod down over the deck. Make sure the rod tip is free of the water if you point it forwards, and prefer a location behind you with the tip pointing towards the rear. Here the rod is less exposed and you can paddle freely. The disadvantage is access. If you suddenly want to grab the rod and fish, finding and freeing it behind you is of course less convenient than in front of you. A strap system with velcro or rubber band will help you.
Vertical rod holders compromise of a vertically mounted tube with a slot for the reel. You set the handle and reel into the tube and strap it down with velcro. The tube(s) can be mounted away from you in the front of the kayak or right behind you. Accessing the rod is usually convenient and quick. Two rods can be mounted pointing each their way.
Do not mount them close and parallel! Fly rods and lines have an amazing ability to tangle when carried close for a long time. And the distance to that tangled tip is even longer when sitting in a kayak.
Other odds and ends
Straps and bungy cords are a very nice addition to the equipment. Straps over the deck or open spaces offer easy attaching of all types of equipment - for a longer duration while you sail or just for a minute or two while you do something else with your hands.
Ice boxes or other types of lidded or open boxes can be very practical. Most standard boxes of a limited size can be fitted on the kayak. Closed kayaks have limited on deck space, while open ones can house even large boxes. Something waterproof is of course preferable.