Thanks!
The clay will "grab" and pull out contaminents that are partially embedded in the paint. Take a look again at those two pictures where I'm holding the clay. All that crud was stuck in the paint. Using a polish or wax just smoothes them out a little but they will still be there. A good way to tell if your paint needs to be clayed is by putting a plastic bag over your hand (like those thin produce bags) and gently going over the paint. If it snags and feels rough, then it's time for the clay bar. Use plently of lube to keep the clay from attempting to smear on the paint. I really like the quick detailer by Meguiar's for that purpose. As the clay starts to show some dirt, just fold it in on itself and mold it again into a small patty that will fit over 3 or 4 or your fingers and keep on going! Remember, claying is really the first step in protecting your paint, at the very least, you still need to wax it. I chose to use the paint cleaner, polish, then the wax for the ultimate in protection.
My process was basically spray a panel (like the fender) with lube and start working the clay back and forth or circular or any of those combinations using gentle pressure. Since the clay is not abrasive, it does not matter how you go over the paint. Spray lots of lube and I keep a microfiber towel on my shoulder to dry the panel immediately after I've done it. Then move onto the next panel.
You can also search YouTube for short videos on how to clay your car. Now, go do it and please post your before and after results!