No one is immune to injury in the kitchen. Cuts and burns especially are common and can be severe. I love cooking with kids and I feel that it's my responsibility to be prepared for potential accidents. Accidents happen in professional kitchens too, and dinner service doesn't stop because the chef cut himself. To be prepared, what items should be found in every kitchen?
The most important things in cooking first aid kits to me are liquid skin, vitamin C powder, and cortisone cream. Obviously antiseptic and bandages are important too, but I assume those are standard.
Liquid skin (basically clear nail polish) is antiseptic and helps to seal up cuts so they stop bleeding faster. It also helps to prevent the effect that any kitchen worker will recognize where an otherwise insignificant cut keeps getting reopened and filling your gloves with blood.
The vitamin C powder is a miracle for burns. You put a pinch in some cold water and rinse the burn with it. It's very soothing and seems to help them heal faster.
Cortisone cream is primarily a bakery thing. Prolonged skin exposure to flour frequently causes rashes and cortisone cream helps to relieve the itching and get the rashes to go away faster.
Equipment isn't really as important as knowing how to use it. Many hiker medics travel with nothing but a half roll of duct tape, gauze, and a couple of salves. Stuff like finger bandages and colored band-aids are really just conveniences more than essentials. In a kitchen you have many of the things you need already without buying anything: running water to cool burns and clean cuts, towels to soak up blood, knives and scissors for cutting gauze, etc.
That being said, conveniences can make things much simpler and easier: it's far faster to get a band-aid than fashion one from medical tape and gauze. So bandages of different shapes are certainly very helpful. Medicated creams or ointments to fight infection on cuts are good. Iodine and medical alcohol also fight infections. Burn creams are generally counter productive, if it's bad enough you need professional help. One of the best things you can have doesn't come in a tube, it's an aloe plant - the best thing for mild burns is first to put under cold water to cool the skin and then apply fresh aloe.
But my first statement is the most important one, the best thing you can do is learn some basics.
Besides the cuts and burns, the other thing that I've made the mistake of a few too many times are not washing my hands immediately or well enough after cutting up hot peppers. (and if you then cut up onions and you wipe the side of your face when you tear up ... and manage to get it in your eyes).
I've done it enough times that I'm surprised that I don't keep an eye-wash bottle in my kitchen ... but I learned a trick from my cousin -- bottled water.
Take the cap off the bottle, then look down and bring the bottle up to your eye ... then squeeze the bottle slightly. You'll want to do this over a sink, as it'll make a mess. It won't have the saline that's in an eyewash, but it'll work in a pinch.
There's another trick that you can use, but it's risky as you've already got at least one potentially unsafe hand. Grab your upper eyeslashes and pull your upper eyelid over the lower one. It'll cause your eye to start producing extra tears. (I also use this when I've gotten something in my eye)
My first-aid kit for burns is an aloe plant. The gel in the leaves really does relieve the pain. And the burn does heal faster. The plant also looks good in the kitchen. The aloe that you find in lotions has been processed so much, that it loses its healing properties.