How to use a gold pan
1. Fill your gold pan 1/2 to 2/3 full of gold bearing material (if it's not in there to begin with, it won't magically appear through panning). If you have a classifier, use it. It will greatly improve your efficiency and gold recovery.
2. Completely submerge pan (and classifier) of material and work it with your hands. Break up any clay balls and/or dirt clods, clean large rocks and wash out root clumps, moss, etc and discard. Gold hides in moss, dirt sticks to muddy rocks, etc, so wash these over your submerged pan.
3. Raise the pan to above or even with the water's surface, keeping your material covered with water, and slurry. The idea here is to turn your pan full of rocks, sand, dirt and water into a liquid. The only way to do that is to use plenty of water and get it all moving at the same time. Some people shake side to side, some use a vigorous circular motion, some keep the fingers of one hand in the mix while going side to side and/or circular and some use a cordless eggbeater (just kidding, but you never know...).
4. Lower your pan so the material is just below the surface of the water and tilted slightly forward. If you have a riffled gold pan, the section with the riffles should be the forward edge of your pan. Rock the pan side to side, up and down, back and forth or in some other motion that lets water flow into the pan and flow back out again. As the water flows out, it will take some of the lighter material with it. Don't dump the material, and don't be afraid of the water. The flow of the water should do all the work.
5. Repeat steps 3 and 4 often. The slurrying stratifies the material and allows the heavy stuff (gold) to settle to the bottom, and the washing removes the light stuff off the top. Steps 3 and 4 are the heart of panning. Repeat often and with plenty of water involved.
6. As you get down under a cup or so of material you may want to introduce a safety. This is a second pan held underwater below your first to catch the material washing out of your pan in case you lose some gold (everyone does). You can then give that material a second panning when you're done with the first.
7. This last bit of material may contain large amounts of black sands which have a tendency to lock up. This makes washing the lighter material off a bit more difficult, but can be countered with a very slight side to side shake while washing. The effect you're going for is more a vibration than anything, just to keep the material from locking up. Be careful here, and bring all the material back to the first riffle frequently to avoid losing gold.
8. Keep repeating steps 3 and 4 (with a little #7 thrown in) until you get down to about a tablespoon of material (not very much is it?). With about a cup of water in your pan (assuming 14" here) tilt the pan gently back so the material is on the higher side, and with a slight shaking and easy-does-it swirl of the water, fan the material out in the bottom of your pan. Most of the gold will show up along the (now) upper edge where it's been sitting since your early slurrying efforts. Gentle shaking and swirling will pull the black sands down away from the gold and a slight tapping on the far edge will make the gold move up against the edge of the pan.
9. Grab your snuffer bottle, squeeze and suck up the gold followed by a little additional water to make sure no gold remains settled in your straw. When you get home, you can remove the straw, replace the lid and transfer the gold to a water filled vial. Again, do this at home, over a gold pan. Stumbling and dropping your vial on some rocks streamside would be a bad thing. Your snuffer bottle won't break and floats if dropped.
Gold Panning Tips
Do your panning in calm, flat water whenever possible, even if you have to trek a bit to get to your material. You want to control the flow of water in and out of your pan, and fast moving water makes that very tricky.
Fine, flat particles of gold can float on the surface tension of water. To avoid floating gold; keep the material wet, do not use hand lotion while panning, use non oily sunscreens and avoid touching your face while panning. A tiny amount (not even a drop) of bio-degradable soap added to the water when doing any required finish panning can also help, as it breaks the surface tension of the water. Some say to use Jet-Dry, but that stuff is just nasty and I wouldn't recommend getting it anywhere near a stream...
Test pan material from several spots before settling in for the day. Gold is not naturally distributed evenly throughout a gold bearing stream, it concentrates based on gravity and water flow at high water (spring floods for example). Some areas will have little to no gold, while a couple feet away may be extremely rich. Without testing, you may be hitting the dead spots and missing the rich.
Always pack out your trash and fill in your holes. Half the fun of prospecting is the natural beauty of the landscape we get to play in, so please do your part to keep it that way.