When doing DIY what safety kit should everyone have (and use)?
Safety glasses and ear plugs (and work gloves too).
Get a good pair of safety glasses so they are comfortable, won't fog up, and do not scratch easily and then take good care of them (just like they are regular glasses). Wearing safety glasses when you are working below anything is huge to not only protect your eyes but also just make it so much easier to do the work. If you are working outside then get the sunglass ones (they can double as just regular sunglasses too - I leave them in my car).
Buy a package of the foam ear plugs... and use them all the time. This is the other item that is huge to protect your hearing but then also make it much much more comfortable to do whatever loud do-it-yourself job you are working on. You will notice a big difference between your stamina when wearing the ear plugs. For instance, if you are pounding in nails, with the ear plugs in you won't notice the sound, but without them, the constant high pitched noise takes a toll on your ears (and wears you out for the job faster). (They also are great for just blocking out noise to let you concentrate.)
The other basic safety equipment is work gloves.
An ABC rated fire extinguisher for each room containing potential fire risks, such as kitchen and garage.
Suitably rated dust mask (respirator). You can breath in some horrible stuff when drilling, cutting, using chemicals, etc.
I just have a basic first aid kit around the house and it has some typical items in it. Bandages, gauze, antiseptic ointment, etc.
I also know where local emergency rooms are just in case it's worse than that. (only had to use it once, knock on wood).
If doing anything electrical, or where electrical wires may be run through your walls, you should have a non-contact voltage tester:
They're about $10-15, and the best money you'll ever spend.
A broom. Clean up your area to avoid tripping/slipping/toe-stubbing/ankle-breaking hazards.
If you wear glasses, or don't like the feel of safety glasses, I recommend a Full Face Shield. I wear glasses and like the extra protection given by my full face shield. The face shield is also easy to flip up out of the way when you are looking at small details or measurements.
I live in a desert, so we have 45 to 60 days where the high is over 100 and 2 or 3 week runs where it is over 110. You have to where the right clothes, inside and out. Outside in the blazing sun, you wear long pants and long shirts, with a hat that has a brim going all the way around to protect your face and neck. And even when the temperature is cooler you still need that protection from the sun. Heat exhaustion hits everywhere so keep a lot of water and take breaks from the sun. Osha closes down jobsites here for no shade and not having X amount of gallons of water per person where there is not running potable water. I know it sound funny and your feet from your house on a diy job but put face, skin, eye, ears, throat and nose, hands and feet in planning.
Another tip is planning the job is having the right tools. Shortcuts can hurt and in the end your job might suffer as well.
Jeff mentioned ear plugs, but if you're working with other people, you'll often have to either shout at each other, which leads to taking out the ear plugs, then you don't put them back in, etc.
Instead, consider 'active earmuffs' -- they're earmuffs that have a microphone and speakers in them. They'll relay the sounds from outside, unless the sound gets too high, and will then shut off the speaker. For some high-pitched noises, they're outside of the sensitivity of the microphone, so you can talk clearer than no ear protection.
The only problems with earmuffs over plugs is that they don't seal quite as well around the head if your eye protection has really thick arms that go over the ears ... but having ear muffs and plugs mean you can use both for really noisy conditions.