The first thing I have to point out is, don't use any of the equipment I'm about to describe unless you have been properly trained in it. And if you haven't been properly trained in it, and nobody else in your group has, do not go white water kayaking. Even a low-grade river can be lethal if you are not experienced - water is incredibly powerful, so a knowledge of whitewater safety and rescue techniques is essential.
1. INDIVIDUAL KIT
OK, firstly and most obviously you will need a kayak. A decent kayak - not one that has a big hole in the bottom with duck tape over it, preferably with air bags in the back, or at least some foam. For running low-grade rivers you will be fine in anything from a basic club boat like a Perception Arc to small playboats like the Wavesport Project which are designed for freestyle (basically doing stunts). On larger volume water you will need a larger kayak - for grade 3-4 a river runner; a medium length, not masively high-volume kayak such as a Liquidlogic Lil Joe will suffice. For grade 4+ to 5 you'll want what's known as a creek boat, a high volume, medium length, hard-as-nails boat that is designed with additional strength to withstand the crushing force of big whitewater. The Pyranha Burn is a good example of a creek boat - and a brilliant one at that.
Then you will need all your usual gear. Make sure you dress appropriately for the time of year - a full dry-suit in midsummer is just as likely to kill you as shorts and t-shirt in winter. You'll need a spray deck, buoyancy aid (or PFD) and helmet whatever the time of year. A lifejacket will not suffice - they are designed to keep your head out of the water even if you are unconscious unlike a PFD, but they make swimming very difficult and will restrict your movement in a kayak as well. Strangely enough, this could cost you your life on a higher grade river. In winter, gloves, wind/waterproof top, wetsuit or dry-suit, thermals and any other warm clothing you can get on you is advisable. You'll also need plenty of food and drink.
A note on helmets - a helmet with ear-protection is ideal, especially on higher grades. Also some helmets can sustain multiple impacts, which is desirable because this is likely to be the case when whitewater kayaking, but these tend to be more expensive. On the other hand, you won't have to keep throwing them away!
In your PFD you should have the following, preferably inside the front pocket if it has one: a waterproof whistle (Fox 40 is a tried and tested make), knife, and emergency rations such as chocolate bars. It doesn't have to be too much of an emergency to eat them - if you are feeling hungry this will lower your performance so keeping stocked up is important. Just remember to replenish your supplies when you get off the water!
This will depend on the classification of the river, but it is a white water river. Within the group you should have at least one first aid kit, repair kit, group shelter, and pair of splits (or paddles that can be taken to pieces in case someone breaks their proper blade), whatever the grade. I have run a class 2-3 rapid without a paddle after being worked in a stopper (an embarrassing mistake really on such a low grade!), and although it worked fine, you wouldn't want to be doing it for long, or on any higher grade.
To be used (as really, for all the equipment in this article) by only those trained and proficient in its usage. A bad rescue can make a dangerous situation far, far worse.
Firstly, a throw bag, preferably of 20m length. These are bags of rope that are generally used for throwing to and rescuing swimmers, but which can also be used for a plethora of other whitewater uses if you are properly trained. Cinches can be used to save the life of a swimmer with their leg trapped, zip-lines can be used to transfer kit reliably from one side of the river to another, and live bait rescues, which involve sending a man into the river to rescue a swimmer, which can be necessary on larger rivers, can all be done using a throw line combined with slings and karabiners. In order to do a live bait rescue it is necessary also to have a chest harness on your PFD: these tend to have an extra buckle underneath the quick release, which will prevent the harness from letting go under even extreme pressure. In general, you do NOT(!) want to be using this; the chest harness is designed to give way at a point just before your ribs break. This will not happen if the buckle is threaded, so the only time this is advisable is if there is a risk of swimming over Death Falls if your harness gives way, in which case a few broken ribs will be a small sacrifice!
Additional PFD kit: sling and karabiners. The sling is loop of tough strap-like material that is used also by climbers, preferebly about two to three metres long, which can be used for white water towing or zip lines and a few other uses (e.g. a rudimentary climbing harness if you're desperate). Karabiners should ideally be screwgate (because you really, REALLY don't want them coming off if you're doing a live bait) and HMS because this is a very strong shape and will allow the krab to move over knots when you are running a zip line with two throw lines tied together.
I think that's about it. Happy paddling!