Panning for Gold is a simple fun thing to do. Sure, some people strike it rich, but it's just fun to do. Here are some basics on how to pan for gold.
The first thing to do would be to buy a gold pan. I've seen these in camping stores and wilderness supply houses. A gold pan is simply a flat broad round pan, sometimes with raised ridges, called riffles inside it. Some gold pans have handles and include screens to filter out larger materials, some don't. Both kinds work just fine, it's just a matter of preference and generally what you can find. A gold pan is shaped kind of like a wok, but slightly flatter.
Next step would be to find where the gold is. There is no use panning for gold in a stream where there is no gold so go to a stream where people have found gold before. Since gold is much heavier than water (about 19 times heavier), the gold stays on the bottom and gets caught in the sand in slow moving areas around bends of the stream and along the shore. It also tends to get stuck in small crevices in rocks and wedged in pieces of wood. Try to find places like this along the stream. Be aware that if it is a stream that is still being actively mined for gold, there may be claims staked along the river. Look for notices posted on trees on any paths to the river, and if someone has filed a claim to that section of river, you can't pan for gold there. Gold miners can be somewhat anti-social and territorial, so this is not a line I would want to cross.
Put about 4 handfuls of material into your gold pan. Submerse the pan in the stream. While holding the gold pan under water, move the pan in a circular motion so that the lighter materials will be carried out of the gold pan. Don't move it too rapidly or you may lose gold along with the rocks and sand. Keep doing this until about half of the material in the gold pan is gone. Lift the pan out of the water and begin swirling it around with it tipped slightly to the side where the riffles are. When all the water is gone dip the pan into the water again, bring it back out, and start swirling again. Keep doing this until nearly all the material in the pan is gone. Occasionally tapping the side of your pan with the heel of your hand will help the heavier gold settle to the bottom of your pan.
With any luck you will now have some small (and I do mean small) flakes of gold in the bottom of your pan. One of the easiest ways to separate the gold from the gravel is to use an eyedropper or a suction pipette. This is easier than using your fingertip, but fingers work well too. A knifeblade or something small and pointed will also come in handy to push those pesky pieces of gravel out of your gold and remove them. Put the gold in a small vial that can be capped.
If you didn't find any gold, don't worry. It takes practice, patience, and a little luck to find, but keep trying. There's nothing like finding gold in the bottom of your gold pan!