BMX clothing can be used as an identification of the rider and as an expression of personality. In the case of identification, if your team has a specific color your clothes would most likely be that of the rest of your team. However you could make your clothing express your personality by choosing your favorite color or a different style of clothing.
If you are planning on racing professionally on dirt tracks I would suggest designing a custom jersey and pants so that you can be identified by your BMX clothing in the midst of a race. If you are being sponsored, most of the time your sponsor will have a team jersey you will be required to wear. If you are just racing for fun or are just a trickster riding in skate parks, all you need is a comfortable shirt that you like. In all cases bright colors are better.
For shoes, if your a racer I would suggest wearing tennis shoes to give better grip on the pedals, but if you are riding in skate parks non-professionally, most people would wear loose fitting skate boarder shoes, that's what I wear.
For your own safety, wearing a helmet is mandatory at most public BMX parks and tracks. I would recommend wearing a helmet if riding any where on your bike. There are two different types of BMX bike helmets.
If you are racing wear a full face helmet. If you are riding non-professionally then either would do.
To supplement your safety you should wear elbow and knee pads also. They could prevent a serious injury and they help keep minor injuries to a minimum.
Wearing gloves can be very useful in the event that your palms get sweaty or you crash because they help improve your grip in the handle bars. Plus if you crash they might keep the skin on your hands.