What basic safety equipment should everyone have?
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.