The Black Rock Desert can get extremely hot and is usually quite sunny. The playa is also very light in color and reflects lots of sunlight. Essentially this means that during most daylight hours on the playa the sun will be bombarding you from every direction. The Playa is about 4,000 feet above sea level and the atmosphere does not filter as much sun - you will burn much faster and easier than you do at the beach. The sun is nothing to trifle with there and a severe sunburn will really make life miserable so pay attention here.
Bring good sun screen and bring way more than you think you need. It takes several ounces to cover an adult body so one bottle just won't cut it. Apply it religiously before you head out each day. Apply it to every square inch of exposed skin - even in places you would not normally be concerned with. It has been said to apply it, then imagine you will be standing on your head all day and apply it to the places you missed.
My personal preference is spray sun screen. It is fast and easy to apply, will get all the nooks and crannies, and will even protect your scalp if sprayed through your hair.
Sun blindness is caused by getting a sunburn on your eyeballs and will really truly fuck up your burn. Bring good sunglasses to protect against it. Having UV rated goggles is also a good idea (or goggles that fit over your sun glasses).
A hat with a brim to shade your eyes is definitely a good idea and will also help prevent sunstroke. Keep in mind that the winds are often intense so you should have some way to strap the hat to your head.
Light loose fitting clothing can help protect you from the sun. For most of us its not essential, however if you burn easily it may be the only way to avoid a sun burn.