We'll start from the head and work our way down. The first thing you should be wearing is a helmet. A beanie or hat might look cool, but a helmet will protect you if you fall. A beanie or hat both have limited safety capabilities.
Next up is the face. You can wear eye protection if you want, but that's not really necessary for skateboarding. You may want to put on a pair of sunglasses if it's a bright day, but that's optional, too.
If you like your teeth, you may want to invest in a mouth guard. This is not necessary, because you're likely to fall on your butt before you fall on your face, but some people like to wear a mouth guard anyway. You can't be too safe, right?
You should wear a long sleeve shirt if you plan on skating without elbow pads. If you feel like a long sleeve shirt restricts your movement, try and find something that fits loose and lets in a breeze.
When you're wearing a short sleeve shirt, it's absolutely a must to wear elbow pads. If you fall forward, it typically goes hands, then elbows. You'll want to keep your elbows protected.
Wrist guards are another optional item that you can think about wearing. If you have weak wrists, or have previously injured a wrist, you may want to pick up a set of wrist guards for yourself.
Wrist guards are great if you're worried about breaking your hand, but a lot of people don't wear them because there's a possibility for upper arm injuries. If you want to prevent your hands from getting scraped up, buy a pair of gloves.
Moving on to the midsection, we have hip and tailbone pads. A lot of people don't even know that these pads exist. New skaters are prone to fall on their tailbones and may want to pick up a tail guard until their skill level improves.
When you skate in shorts, you should wear knee pads. If you can't get a hold of some knee pads for whatever reason, then try to skate in pants. They're the next best thing to knee pads. Again, though, they're not nearly the same, only a good alternative.
The last item on the list is the foot. There a few ankle braces out there, but just like the wrist guards, there are possibilities for upper leg injury. You should probably wear an ankle brace if you have had ankle problems in the past.
It's also necessary to wear the proper shoes while skateboarding. It's not smart to go skateboarding in some loafers or running shoes. Invest in a pair of quality skateboarding shoes. Don't forget that safety comes first!